SSAT Upper Level Math : How to find the area of a right triangle

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

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Example Questions

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Example Question #481 : Geometry

Right Triangle A has hypotenuse 25 inches and one leg of length 24 inches; Right Triangle B has hypotenuse 15 inches and one leg of length 9 inches; Rectangle C has length 16 inches. The area of Rectangle C is the sum of the areas of the two right triangles. What is the width of Rectangle C?

Possible Answers:

Correct answer:

Explanation:

The area of a right triangle is half the product of its legs. In each case, we know the length of one leg and the hypotenuse, so we need to apply the Pythagorean Theorem to find the second leg, then take half the product of the legs:

 

Right Triangle A:

The length of the second leg is

 inches.

The area is 

 square inches.

 

Right Triangle B:

The length of the second leg is

 inches.

The area is 

 square inches.

 

The sum of the areas is  square inches.

 

The area of a rectangle is the product of its length and its height. Therefore, the height is the quotient of the area and the length, which, for Rectangle C, is  inches.

 

Example Question #482 : Geometry

Right Triangle A has legs of lengths 10 inches and 14 inches; Right Triangle B has legs of length 20 inches and 13 inches; Rectangle C has length 30 inches. The area of Rectangle C is the sum of the areas of the two right triangles. What is the height of Rectangle C?

Possible Answers:

Insufficient information is given to determine the height.

Correct answer:

Explanation:

The area of a right triangle is half the product of its legs. The area of Right Triangle A is equal to  square inches; that of Right Triangle B is equal to  square inches. The sum of the areas is  square inches, which is the area of Rectangle C.

 

The area of a rectangle is the product of its length and its height. Therefore, the height is the quotient of the area and the length, which, for Rectangle C, is  inches.

Example Question #483 : Geometry

A right triangle has leg lengths of . What is the area of this triangle?

Possible Answers:

Correct answer:

Explanation:

Since the legs of a right triangle form a right angle, you can use these as the base and the height of the triangle.

Example Question #484 : Geometry

A right triangle has leg lengths of  and . Find the area of the right triangle.

Possible Answers:

Correct answer:

Explanation:

The legs of a right triangle also make up its base and its height.

Example Question #43 : Properties Of Triangles

A right triangle has leg lengths of  and . Find the area of this triangle.

Possible Answers:

Correct answer:

Explanation:

The legs of a right triangle are also its height and its base.

Example Question #485 : Geometry

A right triangle has two legs of lengths  and , respectively. What is the area of the right triangle?

Possible Answers:

Correct answer:

Explanation:

The area  of a right triangle with a base  and a height  can be found with the formula . Since the two legs of a right triangle are perpendicular to each other, we can use these as the base and height in the formula. Therefore:

Example Question #486 : Geometry

A given right triangle has two legs of lengths  and , respectively. What is the area of the triangle?

Possible Answers:

Not enough information to solve

Correct answer:

Explanation:

The area  of a right triangle with a base  and a height  can be found with the formula . Since the two legs of a right triangle are perpendicular to each other, we can use these as the base and height in the formula. Therefore:

Example Question #487 : Geometry

A given right triangle has legs of lengths  and , respectively. What is the area of the right triangle?

Possible Answers:

Not enough information available

Correct answer:

Explanation:

The area  of a right triangle with a base  and a height  can be found with the formula . Since the two legs of a right triangle are perpendicular to each other, we can use these as the base and height in the formula. Therefore:

Example Question #488 : Geometry

The lengths of the hypotenuses of ten similar right triangles form an arithmetic sequence. The smallest triangle has legs of lengths 3 and 4 inches; the second-smallest triangle has a hypotenuse of length one foot.

Which of the following responses comes closest to the area of the largest triangle?

Possible Answers:

8 square feet

6 square feet

5 square feet

9 square feet

7 square feet

Correct answer:

8 square feet

Explanation:

The hypotenuse of the smallest triangle can be calculated using the Pythagorean Theorem:

 inches.

Let  be the lengths of the hypotenuses of the triangles in inches.  and , so their common difference is

The arithmetic sequence formula is 

The length of the hypotenuse of the largest triangle - the tenth triangle - can be found by substituting :

 inches.

The largest triangle has hypotenuse of length 68 inches. Since the triangles are similar, corresponding sides are in proportion. If we let  and  be the lengths of the legs of the largest triangle, then

 

Similarly,

The area of a right triangle is half the product of its legs:

 square inches.

Divide this by 144 to convert to square feet:

Of the given responses, 8 square feet is the closest, and is the correct choice.

 

Example Question #489 : Geometry

Right triangle 3

Figure NOT drawn to scale

In the above figure,  is a right triangle. . What fraction of  has been shaded in?

Possible Answers:

Correct answer:

Explanation:

The length of the leg , which we will call , can be calculated by setting , the length of hypotenuse , and , the length of leg , and applying the Pythagorean Theorem:

Construct the altitude of  - which is also that of , the shaded region - from  to . We call the length of this altitude  (height). The figure is seen below.

Untitled 3

The area of  is one half this height multiplied by the corresponding base length :

The area of , the shaded region - is, similarly,

Therefore, the fraction that  is of  is the fraction of their areas: 

Substituting the measures of the two segments:

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