PSAT Math › Geometry
A right triangle has legs of 15m and 20m. What is the length of the hypotenuse?
30m
45m
35m
40m
25m
The Pythagorean theorem is a2 + b2 = c2, where a and b are legs of the right triangle, and c is the hypotenuse.
(15)2 + (20)2 = c2 so c2 = 625. Take the square root to get c = 25m
All of the following could be the possible side lengths of a triangle EXCEPT:
The length of the third side of a triangle must always be between (but not equal to) the sum and the difference of the other two sides.
For instance, take the example of 2, 6, and 7.
and
. Therefore, the third side length must be greater than 4 and less than 8. Because 7 is greater than 4 and less than 8, it is possible for these to be the side lengths of a triangle.
The 5, 7, 12 answer choice is the only option for which this is not the case.
and
. Therefore, the third side length must be between 2 and 12. Because it is equal to the sum, not less than the sum, it is not possible that these could be the side lengths of a triangle.
A rectangular prism has a volume of 144 and a surface area of 192. If the shortest edge is 3, what is the length of the longest diagonal through the prism?
The volume of a rectangular prism is .
We are told that the shortest edge is 3. Let us call this the height.
We now have , or
.
Now we replace variables by known values:
Now we have:
We have thus determined that the other two edges of the rectangular prism will be 4 and 12. We now need to find the longest diagonal. This is equal to:
If you do not remember how to find this directly, you can also do it in steps. You first find the diagonal across one of the sides (in the plane), by using the Pythagorean Theorem. For example, we choose the side with edges 3 and 4. This diagonal will be:
We then use a plane with one side given by the diagonal we just found (length 5) and the other given by the distance of the 3rd edge (length 12).
This diagonal is then .
We have a square with length 2 sitting in the first quadrant with one corner touching the origin. If the square is inscribed inside a circle, find the equation of the circle.
If the square is inscribed inside the circle, in means the center of the circle is at (1,1). We need to also find the radius of the circle, which happens to be the length from the corner of the square to it's center.
Now use the equation of the circle with the center and .
We get
Refer to the above diagram. .
Give the area of Quadrilateral .
, since both are right; by the Corresponding Angles Theorem,
, and Quadrilateral
is a trapezoid.
By the Angle-Angle Similarity Postulate, since
and
(by reflexivity),
,
and since corresponding sides of similar triangles are in proportion,
, the larger base of the trapozoid;
The smaller base is .
, the height of the trapezoid.
The area of the trapezoid is
Note: Figure NOT drawn to scale.
The above polygon has perimeter 190. Evaluate .
To get the expression equivalent to the perimeter, add the lengths of the sides:
Since the perimeter is 190, we can simplify this to
and solve as follows:
A certain cube has a side length of 25 m. How many square tiles, each with an area of 5 m2, are needed to fully cover the surface of the cube?
100
200
500
750
1000
A cube with a side length of 25m has a surface area of:
25m * 25m * 6 = 3,750 m2
(The surface area of a cube is equal to the area of one face of the cube multiplied by 6 sides. In other words, if the side of a cube is s, then the surface area of the cube is 6_s_2.)
Each square tile has an area of 5 m2.
Therefore, the total number of square tiles needed to fully cover the surface of the cube is:
3,750m2/5m2 = 750
Note: the volume of a cube with side length s is equal to _s_3. Therefore, if asked how many mini-cubes with side length n are needed to fill the original cube, the answer would be:
s3/n3
Refer to the above diagram. .
Give the area of Quadrilateral .
, since both are right; by the Corresponding Angles Theorem,
, and Quadrilateral
is a trapezoid.
By the Angle-Angle Similarity Postulate, since
and
(by reflexivity),
,
and since corresponding sides of similar triangles are in proportion,
, the larger base of the trapozoid;
The smaller base is .
, the height of the trapezoid.
The area of the trapezoid is
The endpoints of a diameter of circle A are located at points and
. What is the area of the circle?
The formula for the area of a circle is given by A =πr2 . The problem gives us the endpoints of the diameter of the circle. Using the distance formula, we can find the length of the diameter. Then, because we know that the radius (r) is half the length of the diameter, we can find the length of r. Finally, we can use the formula A =πr2 to find the area.
The distance formula is
The distance between the endpoints of the diameter of the circle is:
To find the radius, we divide d (the length of the diameter) by two.
Then we substitute the value of r into the formula for the area of a circle.
What is the area of a square that has a diagonal whose endpoints in the coordinate plane are located at (-8, 6) and (2, -4)?
100
100√2
50√2
50
200√2