Right Triangles - PSAT Math
Card 1 of 609
A right triangle has one side equal to 5 and its hypotenuse equal to 14. Its third side is equal to:
A right triangle has one side equal to 5 and its hypotenuse equal to 14. Its third side is equal to:
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The Pythagorean Theorem gives us _a_2 + _b_2 = _c_2 for a right triangle, where c is the hypotenuse and a and b are the smaller sides. Here a is equal to 5 and c is equal to 14, so _b_2 = 142 – 52 = 171. Therefore b is equal to the square root of 171 or approximately 13.07.
The Pythagorean Theorem gives us _a_2 + _b_2 = _c_2 for a right triangle, where c is the hypotenuse and a and b are the smaller sides. Here a is equal to 5 and c is equal to 14, so _b_2 = 142 – 52 = 171. Therefore b is equal to the square root of 171 or approximately 13.07.
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Which of the following could NOT be the lengths of the sides of a right triangle?
Which of the following could NOT be the lengths of the sides of a right triangle?
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We use the Pythagorean Theorem and we calculate that 25 + 49 is not equal to 100.
All of the other answer choices observe the theorem _a_2 + _b_2 = _c_2
We use the Pythagorean Theorem and we calculate that 25 + 49 is not equal to 100.
All of the other answer choices observe the theorem _a_2 + _b_2 = _c_2
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Which set of sides could make a right triangle?
Which set of sides could make a right triangle?
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By virtue of the Pythagorean Theorem, in a right triangle the sum of the squares of the smaller two sides equals the square of the largest side. Only 9, 12, and 15 fit this rule.
By virtue of the Pythagorean Theorem, in a right triangle the sum of the squares of the smaller two sides equals the square of the largest side. Only 9, 12, and 15 fit this rule.
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A right triangle with a base of 12 and hypotenuse of 15 is shown below. Find x.
A right triangle with a base of 12 and hypotenuse of 15 is shown below. Find x.
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Using the Pythagorean Theorem, the height of the right triangle is found to be = √(〖15〗2 –〖12〗2) = 9, so x=9 – 5=4
Using the Pythagorean Theorem, the height of the right triangle is found to be = √(〖15〗2 –〖12〗2) = 9, so x=9 – 5=4
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A right triangle has sides of 36 and 39(hypotenuse). Find the length of the third side
A right triangle has sides of 36 and 39(hypotenuse). Find the length of the third side
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use the pythagorean theorem:
a2 + b2 = c2 ; a and b are sides, c is the hypotenuse
a2 + 1296 = 1521
a2 = 225
a = 15
use the pythagorean theorem:
a2 + b2 = c2 ; a and b are sides, c is the hypotenuse
a2 + 1296 = 1521
a2 = 225
a = 15
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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?
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?
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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.
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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A right triangle has two sides, 9 and x, and a hypotenuse of 15. What is x?
A right triangle has two sides, 9 and x, and a hypotenuse of 15. What is x?
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We can use the Pythagorean Theorem to solve for x.
92 + _x_2 = 152
81 + _x_2 = 225
_x_2 = 144
x = 12
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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The area of a right traingle is 42. One of the legs has a length of 12. What is the length of the other leg?
The area of a right traingle is 42. One of the legs has a length of 12. What is the length of the other leg?
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Area= $\frac{1}{2}$times basetimes height
42=\frac{1}{2}$times basetimes 12
42=6times base
base=7
Area= $\frac{1}{2}$times basetimes height
42=\frac{1}{2}$times basetimes 12
42=6times base
base=7
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If 
and 
, what is the length of 
?
If
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AB is the leg adjacent to Angle A and BC is the leg opposite Angle A.
Since we have a 
triangle, the opposites sides of those angles will be in the ratio 
.
Here, we know the side opposite the sixty degree angle. Thus, we can set that value equal to 
.
which also means
AB is the leg adjacent to Angle A and BC is the leg opposite Angle A.
Since we have a
Here, we know the side opposite the sixty degree angle. Thus, we can set that value equal to
which also means
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Solve for x.
Solve for x.
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Use the Pythagorean Theorem. Let a = 8 and c = 10 (because it is the hypotenuse)
$a^2$$+x^2$$=c^2$
$8^2$$+x^2$$=10^2$
$64+x^2$=100
$x^2$=100-64=36
x=6
Use the Pythagorean Theorem. Let a = 8 and c = 10 (because it is the hypotenuse)
$a^2$$+x^2$$=c^2$
$8^2$$+x^2$$=10^2$
$64+x^2$=100
$x^2$=100-64=36
x=6
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Note: Figure NOT drawn to scale.
Refer to the above diagram. Evaluate 
.
Note: Figure NOT drawn to scale.
Refer to the above diagram. Evaluate
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The altitude perpendicular to the hypotenuse of a right triangle divides that triangle into two smaller triangles similar to each other and the large triangle. Therefore, the sides are in proportion. The hypotenuse of the triangle is equal to
Therefore, we can set up, and solve for 
in, a proportion statement involving the shorter side and hypotenuse of the large triangle and the larger of the two smaller triangles:
The altitude perpendicular to the hypotenuse of a right triangle divides that triangle into two smaller triangles similar to each other and the large triangle. Therefore, the sides are in proportion. The hypotenuse of the triangle is equal to
Therefore, we can set up, and solve for
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In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?
In the figure above, line segments DC and AB are parallel. What is the perimeter of quadrilateral ABCD?
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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.
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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and 
is a right angle.
Which angle or angles must be complementary to 
?
I) 
II) 
III) 
IV) 
V) 
Which angle or angles must be complementary to
I)
II)
III)
IV)
V)
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is a right angle, and, since corresponding angles of similar triangles are congruent, so is 
. A right angle cannot be part of a complementary pair so both can be eliminated.
can be eliminated, since it is congruent to 
; congruent angles are not necessarily complementary.
Since 
is right angle, 
is a right triangle, and 
and 
are its acute angles. That makes 
complementary to 
. Since 
is congruent to 
, it is also complementary to 
.
The correct response is II and V only.
Since
The correct response is II and V only.
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Note: Figures NOT drawn to scale.
Refer to the above figure. Given that 
, evaluate 
.
Note: Figures NOT drawn to scale.
Refer to the above figure. Given that
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By the Pythagorean Theorem, since 
is the hypotenuse of a right triangle with legs 6 and 8, its measure is
.
The similarity ratio of 
to 
is
.
Likewise,
By the Pythagorean Theorem, since
The similarity ratio of
Likewise,
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Note: Figure NOT drawn to scale.
Refer to the above figure. Given that 
, give the area of 
.
Note: Figure NOT drawn to scale.
Refer to the above figure. Given that
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By the Pythagorean Theorem,
The similarity ratio of 
to 
is
,
This can be used to find 
:
The area of 
is therefore
By the Pythagorean Theorem,
The similarity ratio of
This can be used to find
The area of
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Refer to the above figure. Given that , give the perimeter of .
Refer to the above figure. Given that
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By the Pythagorean Theorem,
The similarity ratio of 
to 
is
,
which is subsequently the ratio of the perimeter of 
to that of 
.
The perimeter of 
is
,
so the perimeter of 
can be found using this ratio:
By the Pythagorean Theorem,
The similarity ratio of
which is subsequently the ratio of the perimeter of
The perimeter of
so the perimeter of
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Note: Figure NOT drawn to scale.
Refer to the above diagram. Give the ratio of the perimeter of 
to that of 
.
Note: Figure NOT drawn to scale.
Refer to the above diagram. Give the ratio of the perimeter of
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The altitude of a right triangle from the vertex of its right triangle to its hypotenuse divides it into two similar triangles.
, as the length of the altitude corresponding to the hypotenuse, is the geometric mean of the lengths of the parts of the hypotenuse it forms; that is, it is the square root of the product of the two:
.
The ratio of the smaller sides of these similar triangles is
The ratio of the smaller side of to that of is
or 
: 1,
so this is also the ratio of the perimeter of to that of .
The altitude of a right triangle from the vertex of its right triangle to its hypotenuse divides it into two similar triangles.
The ratio of the smaller sides of these similar triangles is
The ratio of the smaller side of
so this is also the ratio of the perimeter of
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Note: Figure NOT drawn to scale.
Refer to the above diagram. Give the ratio of the perimeter of to that of .
Note: Figure NOT drawn to scale.
Refer to the above diagram. Give the ratio of the perimeter of
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The altitude of a right triangle from the vertex of its right triangle to its hypotenuse divides it into two similar triangles.
, as the length of the altitude corresponding to the hypotenuse, is the geometric mean of the lengths of the parts of the hypotenuse it forms; that is, it is the square root of the product of the two:
.
The ratio of the smaller sides of these similar triangles is
The ratio of the smaller side of to that of is
or : 1,
so this is also the ratio of the perimeter of to that of .
The altitude of a right triangle from the vertex of its right triangle to its hypotenuse divides it into two similar triangles.
The ratio of the smaller sides of these similar triangles is
The ratio of the smaller side of
so this is also the ratio of the perimeter of
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Acute angles x and y are inside a right triangle. If x is four less than one third of 21, what is y?
Acute angles x and y are inside a right triangle. If x is four less than one third of 21, what is y?
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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.
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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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?
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?
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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.
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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