SSAT Upper Level Math : Properties of Triangles

Study concepts, example questions & explanations for SSAT Upper Level Math

varsity tutors app store varsity tutors android store

Example Questions

Example Question #3 : Perimeter Of A Triangle

What is the perimeter of a right triangle with hypotenuse \displaystyle 91 and a leg of length \displaystyle 35?

Possible Answers:

It cannot be determined from the information given.

\displaystyle 175

\displaystyle 119

\displaystyle 126

\displaystyle 210

Correct answer:

\displaystyle 210

Explanation:

Using the Pythagorean Theorem, the length of the second leg can be determined.

\displaystyle a^2+b^2=c^2

We are given the length of the hypotenuse and one leg.

\displaystyle a=35, c=91

\displaystyle b^2=c^2-a^2\rightarrow b=\sqrt{c^2-a^2}

\displaystyle \sqrt{91^{2}-35^{2}} = \sqrt{8,281-1,225} = \sqrt{7,056} = 84

The perimeter of the triangle is the sum of the lengths of the sides.

\displaystyle P=35+84+91 = 210

Example Question #1 : Properties Of Triangles

Which of these polygons has the same perimeter as a right triangle with legs 6 feet and 8 feet?

Possible Answers:

A regular decagon with sidelength one yard.

A regular hexagon with sidelength one yard.

A regular pentagon with sidelength one yard.

None of the other responses is correct.

A regular octagon with sidelength one yard.

Correct answer:

A regular octagon with sidelength one yard.

Explanation:

A right triangle with legs 6 feet and 8 feet has hypotentuse 10 feet, as this is a right triangle that confirms to the well-known Pythagorean triple 6-8-10. The perimeter is therefore \displaystyle 6 + 8 + 10 = 24 feet, or 8 yards.

We are looking for a polygon with this perimeter. Each choice is a polygon with all sides one yard long, so we want the polygon with eight sides - the regular octagon is the correct choice.

Example Question #442 : Geometry

The lengths of the legs of a right triangle are \displaystyle 14 units and \displaystyle 48 units. What is the perimeter of this right triangle?

Possible Answers:

\displaystyle 92 units

\displaystyle 42 units

\displaystyle 112 units

\displaystyle 85 units

Correct answer:

\displaystyle 112 units

Explanation:

First, we need to use the Pythagorean Theorem to find the hypotenuse of the triangle.

\displaystyle 14^2+48^2=\text{Hypotenuse}^2

\displaystyle 196+2304=\text{Hypotenuse}^2

\displaystyle \text{Hypotenuse}^2=2500

\displaystyle \text{Hypotenuse}=\sqrt{2500}=50

Now, add up all three side lengths to find the perimeter of the triangle.

\displaystyle 50+14+48=112

Example Question #3 : How To Find The Perimeter Of A Right Triangle

A right triangle has leg lengths of \displaystyle 7\:cm and \displaystyle 8\:cm. Find the perimeter of this triangle.

Possible Answers:

\displaystyle 15\:cm

\displaystyle 15+\sqrt{113}\:cm

\displaystyle 164\:cm

\displaystyle \sqrt{113}\:cm

Correct answer:

\displaystyle 15+\sqrt{113}\:cm

Explanation:

First, use the Pythagorean Theorem to find the length of the hypotenuse.

\displaystyle a^2+b^2=c^2

Substituting in \displaystyle 7 and \displaystyle 8 for \displaystyle a and \displaystyle b (the lengths of the triangle's legs), we get:

\displaystyle 7^2+8^2=c^2

\displaystyle 49+64=c^2

\displaystyle 113=c^2

\displaystyle \sqrt{113}=\sqrt{c^2}

\displaystyle \sqrt{113}=c

Now, add up the three sides to find the perimeter:

\displaystyle 7+8+\sqrt{113}=15+\sqrt{113}

Example Question #4 : How To Find The Perimeter Of A Right Triangle

What is the perimeter of a right triangle with legs of length \displaystyle 5 and \displaystyle 7, respectively?

Possible Answers:

\displaystyle 12+\sqrt{74}

\displaystyle 5+\sqrt{74}

\displaystyle 7+\sqrt{74}

\displaystyle \sqrt{74}

\displaystyle 12

Correct answer:

\displaystyle 12+\sqrt{74}

Explanation:

In order to find the perimeter \displaystyle P of the right triangle, we need to first find the missing length of the hypotenuse. In order to find the hypotenuse, use the Pythagorean Theorem:

\displaystyle a^{2}+ b^{2}=c^{2}, where \displaystyle a and \displaystyle b are the lengths of the legs of the triangle, and \displaystyle c is the length of the hypotenuse.

Substituting in our known values:

\displaystyle 5^{2}+ 7^{2}=c^{2}

\displaystyle 25+ 49=c^{2}

\displaystyle 74=c^{2}

\displaystyle \sqrt{74}=c

Now that we have the lengths of all sides of the right triangle, we can calculate the perimeter:

\displaystyle P=5+7+\sqrt{74}

\displaystyle P=12+\sqrt{74}

Example Question #5 : How To Find The Perimeter Of A Right Triangle

What is the perimeter of a right triangle with a hypotenuse of length \displaystyle 14 and a leg of length \displaystyle 8?

Possible Answers:

\displaystyle 8+ 2\sqrt{33}

\displaystyle 22+ 2\sqrt{33}

Not enough information provided

\displaystyle 2\sqrt{33}

\displaystyle 14+ 2\sqrt{33}

Correct answer:

\displaystyle 22+ 2\sqrt{33}

Explanation:

In order to find the perimeter \displaystyle P of the right triangle, we need to first find the missing length of the second leg. In order to find the second leg, use the Pythagorean Theorem:

\displaystyle a^{2}+ b^{2}=c^{2}, where \displaystyle a and \displaystyle b are the lengths of the legs of the triangle, and \displaystyle c is the length of the hypotenuse.

Substituting in our known values:

\displaystyle 8^{2}+ b^{2}=14^{2}

Subtracting \displaystyle 8^2 from each side of the equation lets us isolate the variable for which we are solving:

\displaystyle b^{2}=14^{2}-8^{2}

\displaystyle b^{2}=196-64

\displaystyle b^{2}=132

\displaystyle b=\sqrt{132}

\displaystyle b=\sqrt{33\times 4}

\displaystyle b=2\sqrt{33}

Now that we have the lengths of all three sides of the right triangle, we can calculate the perimeter \displaystyle P:

\displaystyle P=14+8+2\sqrt{33}

\displaystyle P=22+2\sqrt{33}

Example Question #6 : How To Find The Perimeter Of A Right Triangle

Find the perimeter \displaystyle P of a right triangle with two legs of length \displaystyle 8 and \displaystyle 12, respectively. 

Possible Answers:

\displaystyle 8+4\sqrt{13}

\displaystyle 12+4\sqrt{13}

Not enough information provided 

\displaystyle 20+4\sqrt{13}

\displaystyle 4\sqrt{13}

Correct answer:

\displaystyle 20+4\sqrt{13}

Explanation:

In order to find the perimeter \displaystyle P of the right triangle, we need to first find the missing length of the hypotenuse. In order to find the length of the hypotenuse, use the Pythagorean Theorem:

\displaystyle a^{2}+ b^{2}=c^{2}, where \displaystyle a and \displaystyle b are the lengths of the legs of the triangle, and \displaystyle c is the length of the hypotenuse.

Substituting in our known values:

\displaystyle 8^{2}+ 12^{2}=c^{2}

\displaystyle 64+144=c^{2}

\displaystyle 208=c^{2}

\displaystyle \sqrt{208}=c

\displaystyle \sqrt{13\times 16}=c

\displaystyle 4\sqrt{13}=c

Now that we have all three sides of the right triangle, we can calculate the perimeter:

\displaystyle P=8+12+4\sqrt{13}

\displaystyle P=20+4\sqrt{13}

Example Question #1 : Right Triangles

If a \displaystyle 3-4-5 right triangle is similar to a \displaystyle 6-8-10 right triangle, which of the other triangles must also be a similar triangle?

Possible Answers:

\displaystyle \frac{1}{6}-\frac{1}{8}-\frac{1}{10}

\displaystyle \frac{1}{3}-\frac{1}{4}-\frac{1}{5}

\displaystyle 12-16-20

\displaystyle 9-10-11

\displaystyle 4-5-6

Correct answer:

\displaystyle 12-16-20

Explanation:

For the triangles to be similar, the dimensions of all sides must have the same ratio by dividing the 3-4-5 triangle.

The 6-8-10 triangle will have a scale factor of 2 since all dimensions are doubled the original 3-4-5 triangle.

The only correct answer that will yield similar ratios is the  \displaystyle 12-16-20 triangle with a scale factor of 4 from the 3-4-5 triangle.  

The other answers will yield different ratios.

Example Question #2 : Right Triangles

What is the main difference between a right triangle and an isosceles triangle? 

Possible Answers:

A right triangle has to have a \displaystyle 90^{\circ} angle and an isosceles triangle has to have \displaystyle 2 equal, base angles. 

A right triangle has to have a \displaystyle 50^{\circ} angle and an isosceles triangle has to have \displaystyle 2 equal, base angles. 

A right triangle has to have a \displaystyle 90^{\circ} angle and an isosceles triangle has to have \displaystyle 3equal, base angles. 

A right triangle has to have a \displaystyle 90^{\circ} angle and an isosceles triangle has to have \displaystyle 4 equal, base angles. 

An isosceles triangle has to have a \displaystyle 90^{\circ} angle and a right triangle has to have \displaystyle 2 equal, base angles. 

Correct answer:

A right triangle has to have a \displaystyle 90^{\circ} angle and an isosceles triangle has to have \displaystyle 2 equal, base angles. 

Explanation:

By definition, a right triangle has to have one right angle, or a \displaystyle 90^{\circ} angle, and an isosceles triangle has \displaystyle 2 equal base angles and two equal side lengths. 

Example Question #3 : Properties Of Triangles

A right triangle has a hypotenuse of 39 and one leg is 36. What is the length of the other leg?

Possible Answers:

\displaystyle 3

\displaystyle 27

\displaystyle 19

\displaystyle 33

\displaystyle 15

Correct answer:

\displaystyle 15

Explanation:

You may recognize these numbers as multiples of 13 and 12 (each by a factor of 3) and remember that sides of length 5, 12 and 13 make a special right triangle. So the other leg would be 15 \displaystyle (5\times 3).

If you don't remember this, you can use Pythagorean theorem:

\displaystyle 39^{2}-36^{2}= 1521-1296=225

\displaystyle \sqrt{225}=15

Learning Tools by Varsity Tutors