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SAT Math : How to find the perimeter of a right triangle

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Example Question #1 : How To Find The Perimeter Of A Right Triangle

Three points in the xy-coordinate system form a triangle.

The points are (-1,5) (-1,1) (4,5) .

What is the perimeter of the triangle?

Possible Answers:

$9 + \sqrt{26}$

$9 + \sqrt{41}$

$9 + \sqrt{71}$

Correct answer:

$9 + \sqrt{41}$

Explanation:

Drawing points gives sides of a right triangle of 4, 5, and an unknown hypotenuse.

Using the pythagorean theorem we find that the hypotenuse is $\sqrt{41}$ .

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Example Question #11 : Triangles

Triangle

Based on the information given above, what is the perimeter of triangle ABC?

Possible Answers:

$35+17.5\sqrt{2}$

$17.5+17.5\sqrt{3}$

$52.5\sqrt{3}$

$306.25\sqrt{3}$

$52.5+17.5\sqrt{3}$

Correct answer:

$52.5+17.5\sqrt{3}$

Explanation:

Triangle-solution

Consult the diagram above while reading the solution. Because of what we know about supplementary angles, we can fill in the inner values of the triangle. Angles A and B can be found by the following reductions:

A + 120 = 180; A = 60

B + 150 = 180; B = 30

Since we know A + B + C = 180 and have the values of A and B, we know:

60 + 30 + C = 180; C = 90

This gives us a 30:60:90 triangle. Now, since 17.5 is across from the 30° angle, we know that the other two sides will have to be √3 and 2 times 17.5; therefore, our perimeter will be as follows:

$17.5+2(17.5)+17.5\sqrt{3}=52.2+17.5\sqrt{3}$

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Example Question #43 : Geometry

Right triangle 7

Give the perimeter of the provided triangle.

Possible Answers:

$24+ 12 \sqrt{3}$

$24+ 12 \sqrt{5}$

$24+ 12\sqrt{2}$

Correct answer:

$24+ 12\sqrt{2}$

Explanation:

The figure shows a right triangle. The acute angles of a right triangle have measures whose sum is $90^{\circ }$ , so

$m \angle A+ m \angle C = 90^{\circ }$

Substituting $45^{\circ }$ for $m \angle C$ :

$m \angle A+ 45^{\circ } = 90^{\circ }$

$m \angle A= 45^{\circ }$

This makes $\bigtriangleup ABC$ a 45-45-90 triangle.

By the 45-45-90 Triangle Theorem, legs $\overline{AB}$ and $\overline{BC}$ are of the same length, so

BC = AB = 12 .

Also by the 45-45-90 Triangle Theorem, the length of hypotenuse $\overline{AC}$ is equal to that of leg $\overline{BC}$ multiplied by $\sqrt{2}$ . Therefore,

$AC = AB \cdot \sqrt{2} = 12 \sqrt{2}$ .

The perimeter of the triangle is

$P = AB + B C + AC = 12+ 12+ 12 \sqrt{2} = 24 + 12 \sqrt{2}$

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Example Question #44 : Geometry

Right triangle 7

What is the perimeter of the triangle above?

Possible Answers:

$12+12 \sqrt{2}$

$12+ 12\sqrt{3}$

$12+ 8\sqrt{3}$

Correct answer:

$12+12 \sqrt{2}$

Explanation:

The figure shows a right triangle. The acute angles of a right triangle have measures whose sum is $90^{\circ }$ , so

$m \angle A+ m \angle C = 90^{\circ }$

Substituting $45^{\circ }$ for $m \angle C$ :

$m \angle A+ 45^{\circ } = 90^{\circ }$

$m \angle A= 45^{\circ }$

This makes $\bigtriangleup ABC$ a 45-45-90 triangle. By the 45-45-90 Triangle Theorem, the length of leg $\overline{BC}$ is equal to that of hypotenuse $\overline{AC}$ , the length of which is 12, divided by $\sqrt{2}$ . Therefore,

$BC = \frac{AC}{\sqrt{2}} = \frac{12}{\sqrt{2}}$

Rationalize the denominator by multiplying both halves of the fraction by $\sqrt{2}$ :

$BC = \frac{12 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{12 \sqrt{2}}{2} = 6 \sqrt{2}$

By the same reasoning, $AB= 6 \sqrt{2}$ .

The perimeter of the triangle is

P = AB + B C + AC

$P= 6 \sqrt{2} + 6 \sqrt{2} + 12 = 12+ 12 \sqrt{2}$

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SAT Math : How to find the perimeter of a right triangle

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #1 : How To Find The Perimeter Of A Right Triangle

Example Question #11 : Triangles

Example Question #43 : Geometry

Example Question #44 : Geometry

All SAT Math Resources

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