How to find the area of a right triangle - SSAT Upper Level Quantitative
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What is the area of a right triangle whose hypotenuse is 13 inches and whose legs each measure a number of inches equal to an integer?
What is the area of a right triangle whose hypotenuse is 13 inches and whose legs each measure a number of inches equal to an integer?
Tap to reveal answer
We are looking for a Pythagorean triple - that is, three integers that satisfy the relationship 
. We know that 
, and the only Pythagorean triple with 
is 
. The legs of the triangle are therefore 5 and 12, and the area of the right triangle is
We are looking for a Pythagorean triple - that is, three integers that satisfy the relationship
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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?
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?
Tap to reveal answer
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.
The area of a right triangle is half the product of its legs. The area of Right Triangle A is equal to
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
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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?
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?
Tap to reveal answer
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.
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
The area is
Right Triangle B:
The length of the second leg is
The area is
The sum of the areas is
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
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A right triangle has leg lengths of 
. What is the area of this triangle?
A right triangle has leg lengths of
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Since the legs of a right triangle form a right angle, you can use these as the base and the height of the triangle.
Since the legs of a right triangle form a right angle, you can use these as the base and the height of the triangle.
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A right triangle has leg lengths of 
and 
. Find the area of the right triangle.
A right triangle has leg lengths of
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The legs of a right triangle also make up its base and its height.
The legs of a right triangle also make up its base and its height.
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A right triangle has leg lengths of 
and 
. Find the area of this triangle.
A right triangle has leg lengths of
Tap to reveal answer
The legs of a right triangle are also its height and its base.
The legs of a right triangle are also its height and its base.
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A given right triangle has two legs of lengths 
and 
, respectively. What is the area of the triangle?
A given right triangle has two legs of lengths
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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:
The area
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A right triangle has two legs of lengths 
and 
, respectively. What is the area of the right triangle?
A right triangle has two legs of lengths
Tap to reveal answer
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:
The area
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A given right triangle has legs of lengths 
and 
, respectively. What is the area of the right triangle?
A given right triangle has legs of lengths
Tap to reveal answer
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:
The area
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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?
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?
Tap to reveal answer
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.
The hypotenuse of the smallest triangle can be calculated using the Pythagorean Theorem:
Let
The arithmetic sequence formula is
The length of the hypotenuse of the largest triangle - the tenth triangle - can be found by substituting
The largest triangle has hypotenuse of length 68 inches. Since the triangles are similar, corresponding sides are in proportion. If we let
Similarly,
The area of a right triangle is half the product of its legs:
Divide this by 144 to convert to square feet:
Of the given responses, 8 square feet is the closest, and is the correct choice.
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Figure NOT drawn to scale
In the above figure, 
is a right triangle. 
, 
, 
. What fraction of 
has been shaded in?
Figure NOT drawn to scale
In the above figure,
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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.
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:
The length of the leg
Construct the altitude of
The area of
The area of
Therefore, the fraction that
Substituting the measures of the two segments:
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What is the area of a right triangle whose hypotenuse is 13 inches and whose legs each measure a number of inches equal to an integer?
What is the area of a right triangle whose hypotenuse is 13 inches and whose legs each measure a number of inches equal to an integer?
Tap to reveal answer
We are looking for a Pythagorean triple - that is, three integers that satisfy the relationship . We know that , and the only Pythagorean triple with is . The legs of the triangle are therefore 5 and 12, and the area of the right triangle is
We are looking for a Pythagorean triple - that is, three integers that satisfy the relationship
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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?
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?
Tap to reveal answer
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.
The area of a right triangle is half the product of its legs. The area of Right Triangle A is equal to
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
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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?
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?
Tap to reveal answer
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.
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
The area is
Right Triangle B:
The length of the second leg is
The area is
The sum of the areas is
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
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A right triangle has leg lengths of . What is the area of this triangle?
A right triangle has leg lengths of
Tap to reveal answer
Since the legs of a right triangle form a right angle, you can use these as the base and the height of the triangle.
Since the legs of a right triangle form a right angle, you can use these as the base and the height of the triangle.
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A right triangle has leg lengths of and . Find the area of the right triangle.
A right triangle has leg lengths of
Tap to reveal answer
The legs of a right triangle also make up its base and its height.
The legs of a right triangle also make up its base and its height.
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A right triangle has leg lengths of and . Find the area of this triangle.
A right triangle has leg lengths of
Tap to reveal answer
The legs of a right triangle are also its height and its base.
The legs of a right triangle are also its height and its base.
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A given right triangle has two legs of lengths and , respectively. What is the area of the triangle?
A given right triangle has two legs of lengths
Tap to reveal answer
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:
The area
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A right triangle has two legs of lengths and , respectively. What is the area of the right triangle?
A right triangle has two legs of lengths
Tap to reveal answer
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:
The area
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A given right triangle has legs of lengths and , respectively. What is the area of the right triangle?
A given right triangle has legs of lengths
Tap to reveal answer
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:
The area
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