Right Triangles - PSAT Math

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Question

The ratio for the side lengths of a right triangle is 3:4:5. If the perimeter is 48, what is the area of the triangle?

Answer

We can model the side lengths of the triangle as 3x, 4x, and 5x. We know that perimeter is 3x+4x+5x=48, which implies that x=4. This tells us that the legs of the right triangle are 3x=12 and 4x=16, therefore the area is A=1/2 bh=(1/2)(12)(16)=96.

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Question

Jim leaves his home and walks 10 minutes due west and 5 minutes due south. If Jim could walk a straight line from his current position back to his house, how far, in minutes, is Jim from home?

Answer

By using Pythagorean Theorem, we can solve for the distance “as the crow flies” from Jim to his home:

102 + 52 = _x_2

100 + 25 = _x_2

√125 = x, but we still need to factor the square root

√125 = √25*5, and since the √25 = 5, we can move that outside of the radical, so

5√5= x

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Question

If one of the short sides of a 45-45-90 triangle equals 5, how long is the hypotenuse?

Answer

Using the Pythagorean theorem, _x_2 + _y_2 = _h_2. And since it is a 45-45-90 triangle the two short sides are equal. Therefore 52 + 52 = _h_2 . Multiplied out 25 + 25 = _h_2.

Therefore _h_2 = 50, so h = √50 = √2 * √25 or 5√2.

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Question

If the length of CB is 6 and the angle C measures 45º, what is the length of AC in the given right triangle?

Answer

Pythagorean Theorum

AB2 + BC2 = AC2

If C is 45º then A is 45º, therefore AB = BC

AB2 + BC2 = AC2

62 + 62 = AC2

262 = AC2

AC = √(262) = 6√2

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Question

A square enclosure has a total area of 3,600 square feet. What is the length, in feet, of a diagonal across the field rounded to the nearest whole number?

Answer

In order to find the length of the diagonal accross a square, we must first find the lengths of the individual sides.

The area of a square is found by multiply the lengths of 2 sides of a square by itself.

So, the square root of 3,600 comes out to 60 ft.

The diagonal of a square can be found by treating it like a right triangle, and so, we can use the pythagorean theorem for a right triangle.

602 + 602 = C2

the square root of 7,200 is 84.8, which can be rounded to 85

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Question

Dan drives 5 miles north and then 8 miles west to get to school. If he walks, he can take a direct path from his house to the school, cutting down the distance. How long is the path from Dan's house to his school?

Answer

We are really looking for the hypotenuse of a triangle that has legs of 5 miles and 8 miles.

Apply the Pythagorean Theorem:

a2 + b2 = c2

25 + 64 = c2

89 = c2

c = 9.43 miles

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Question

Bob the Helicopter is at 30,000 ft. above sea level, and as viewed on a map his airport is 40,000 ft. away. If Bob travels in a straight line to his airport at 250 feet per second, how many minutes will it take him to arrive?

Answer

Draw a right triangle with a height of 30,000 ft. and a base of 40,000 ft. The hypotenuse, or distance travelled, is then 50,000ft using the Pythagorean Theorem. Then dividing distance by speed will give us time, which is 200 seconds, or 3 minutes and 20 seconds.

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Question

A right triangle has two sides, 9 and x, and a hypotenuse of 15. What is x?

Answer

We can use the Pythagorean Theorem to solve for x.

92 + _x_2 = 152

81 + _x_2 = 225

_x_2 = 144

x = 12

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Question

To get from his house to the hardware store, Bob must drive 3 miles to the east and then 4 miles to the north. If Bob was able to drive along a straight line directly connecting his house to the store, how far would he have to travel then?

Answer

Since east and north directions are perpendicular, the possible routes Bob can take can be represented by a right triangle with sides a and b of length 3 miles and 5 miles, respectively. The hypotenuse c represents the straight line connecting his house to the store, and its length can be found using the Pythagorean theorem: _c_2 = 32+ 42 = 25. Since the square root of 25 is 5, the length of the hypotenuse is 5 miles.

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Question

A right triangle has legs of 15m and 20m. What is the length of the hypotenuse?

Answer

The Pythagorean theorem is a2 + b2 = c2, where a and b are legs of the right triangle, and c is the hypotenuse.

(15)2 + (20)2 = c2 so c2 = 625. Take the square root to get c = 25m

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Question

Acute angles x and y are inside a right triangle. If x is four less than one third of 21, what is y?

Answer

We know that the sum of all the angles must be 180 and we already know one angle is 90, leaving the sum of x and y to be 90.

Solve for x to find y.

One third of 21 is 7. Four less than 7 is 3. So if angle x is 3 then that leaves 87 for angle y.

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Question

If a right triangle has one leg with a length of 4 and a hypotenuse with a length of 8, what is the measure of the angle between the hypotenuse and its other leg?

Answer

The first thing to notice is that this is a 30o:60o:90o triangle. If you draw a diagram, it is easier to see that the angle that is asked for corresponds to the side with a length of 4. This will be the smallest angle. The correct answer is 30.

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Question

In the figure above, what is the positive difference, in degrees, between the measures of angle ACB and angle CBD?

Answer

In the figure above, angle ADB is a right angle. Because side AC is a straight line, angle CDB must also be a right angle.

Let’s examine triangle ADB. The sum of the measures of the three angles must be 180 degrees, and we know that angle ADB must be 90 degrees, since it is a right angle. We can now set up the following equation.

x + y + 90 = 180

Subtract 90 from both sides.

x + y = 90

Next, we will look at triangle CDB. We know that angle CDB is also 90 degrees, so we will write the following equation:

y – 10 + 2_x_ – 20 + 90 = 180

y + 2_x_ + 60 = 180

Subtract 60 from both sides.

y + 2_x_ = 120

We have a system of equations consisting of x + y = 90 and y + 2_x_ = 120. We can solve this system by solving one equation in terms of x and then substituting this value into the second equation. Let’s solve for y in the equation x + y = 90.

x + y = 90

Subtract x from both sides.

y = 90 – x

Next, we can substitute 90 – x into the equation y + 2_x_ = 120.

(90 – x) + 2_x_ = 120

90 + x = 120

x = 120 – 90 = 30

x = 30

Since y = 90 – x, y = 90 – 30 = 60.

The question ultimately asks us to find the positive difference between the measures of ACB and CBD. The measure of ACB = 2_x_ – 20 = 2(30) – 20 = 40 degrees. The measure of CBD = y – 10 = 60 – 10 = 50 degrees. The positive difference between 50 degrees and 40 degrees is 10.

The answer is 10.

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Question

Which of the following sets of line-segment lengths can form a triangle?

Answer

In any given triangle, the sum of any two sides is greater than the third. The incorrect answers have the sum of two sides equal to the third.

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Question

In right $\Delta ABC$ , $\angle ABC$ $=$ $2x$ and $\angle BCA= \frac{x}{2}$ .

What is the value of $x$ ?

Answer

There are 180 degrees in every triangle. Since this triangle is a right triangle, one of the angles measures 90 degrees.

Therefore, $90 + 2x + \frac{x}{2}= 180$ .

$90=2.5x$

$x=36$

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Question

Refer to the above diagram. Which of the following gives a valid alternative name for $\Delta AGF$ ?

Answer

A triangle can be named after its three vertices in any order, so all of the choices given are valid.

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Question

You are given triangles $\Delta MNO$ and $\Delta PQR$ ,with $\angle N$ and $\angle Q$ both right angles, and . Which of these statements, along with what you are given, is not enough to prove that $\Delta MNO \cong \Delta PQR$ ?

I)

II) $\angle O \cong \angle R$

III) $\Delta MNO$ and $\Delta PQR$ have the same area.

Answer

, and the right angles are $\angle N$ and $\angle Q$ , so we have two right triangles with congruent legs.

If we also know that , then the hypotenuses of the right triangles are also congruent, and this sets up the conditions of the Hypotenuse-Leg Theorem.

If we also know that $\angle O \cong \angle R$ , then, along with the fact that $\angle N \cong \angle Q$ (both being right angles) and nonincluded sides , the conditions of the Angle-Angle-Side Theorem are set up.

If we also know $\Delta MNO$ and $\Delta PQR$ have the same area, we can demonstrate that the other legs are congruent. The area of a right triangle is half the product of its legs, and since we have the same areas,

$\frac{1}{2} (MN)(NO) = \frac{1}{2} (PQ)(QR)$

$2 \times \frac{1}{2} (MN)(NO) =2 \times \frac{1}{2} (PQ)(QR)$

Since ,

$\frac{(MN)(NO)}{MN} = \frac{(PQ)(QR)}{PQ}$

The legs and the included angles (the right angles) are congruent, thus setting up the conditions for the Angle-Side-Angle Postulate.

In all three cases, congruence follows, so the correct response is "Any of the three statements is enough to prove congruence."

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Question

In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?

Answer

Because DC and AB are parallel, this means that angles CDB and ABD are equal. When two parallel lines are cut by a transversal line, alternate interior angles (such as CDB and ABD) are congruent.

Now, we can show that triangles ABD and BDC are similar. Both ABD and BDC are right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that angles CDB and ABD are congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus triangles ABD and BDC are similar triangles.

We can use the similarity between triangles ABD and BDC to find the lengths of BC and CD. The length of BC is proportional to the length of AD, and the length of CD is proportional to the length of DB, because these sides correspond.

We don’t know the length of DB, but we can find it using the Pythagorean Theorem. Let a, b, and c represent the lengths of AD, AB, and BD respectively. According to the Pythagorean Theorem:

_a_2 + _b_2 = _c_2

152 + 202 = _c_2

625 = _c_2

c = 25

The length of BD is 25.

We now have what we need to find the perimeter of the quadrilateral.

Perimeter = sum of the lengths of AB, BC, CD, and DA.

Perimeter = 20 + 18.75 + 31.25 + 15 = 85

The answer is 85.

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Question

$\Delta GHJ \sim \Delta KLM$ and $\angle G$ is a right angle.

Which angle or angles must be complementary to $\angle H$ ?

I) $\angle G$

II) $\angle J$

III) $\angle K$

IV) $\angle L$

V) $\angle M$

Answer

$\angle G$ is a right angle, and, since corresponding angles of similar triangles are congruent, so is $\angle K$ . A right angle cannot be part of a complementary pair so both can be eliminated.

$\angle L$ can be eliminated, since it is congruent to $\angle H$ ; congruent angles are not necessarily complementary.

Since $\angle G$ is right angle, $\Delta GHJ$ is a right triangle, and $\angle H$ and $\angle J$ are its acute angles. That makes $\angle J$ complementary to $\angle H$ . Since $\angle M$ is congruent to $\angle J$ , it is also complementary to $\angle H$ .

The correct response is II and V only.

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Question

Note: Figures NOT drawn to scale.

Refer to the above figure. Given that $\Delta ABC \sim \Delta DE F$ , evaluate .

Answer

By the Pythagorean Theorem, since $\overline{AC}$ is the hypotenuse of a right triangle with legs 6 and 8, its measure is

$AC = \sqrt{6^{2} + 8 ^{2}} = \sqrt{36 + 64 } = \sqrt{100} = 10$ .

The similarity ratio of $\Delta DE F$ to $\Delta ABC$ is

$\frac{DF}{AC} = \frac{70}{10} = 7$ .

Likewise,

$\frac{DE}{AB} = 7$

$\frac{DE}{8} = 7$

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