Answer these questions. Please show your work.
1. Right triangle has an angle that is 58deg…what is the measure of the 3rd angle?
2. A triangle has sides of 10, 9, 13 cm. Can it be a right triangle? Prove
3. Find the distance between points (-2,3) and (4,1). round to nearest tenth
4. Find the diagonal of a rectangular prism box that has measures of 3 x 4 x 10 cm. Round to nearest tenth.
5. A circus tent is the shape of a square pyramid. It has a base edge of 20meter, a slant height of 13.5m. Find the height of the tent. Round to nearest tenth.
1. The sum of the angles in a triangle is always 180 degrees. Since the right triangle has a 90 degree angle, the sum of the other two angles must be 180 - 90 = 90 degrees. Therefore, the measure of the 3rd angle is 90 degrees.
2. To determine if the triangle is a right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the longest side (hypotenuse).
Let's check if the given triangle satisfies this theorem:
10^2 + 9^2 = 100 + 81 = 181
13^2 = 169
Since 181 is not equal to 169, the triangle does not satisfy the Pythagorean theorem and it is not a right triangle.
3. The distance between two points can be found using the distance formula:
√((x2 - x1)^2 + (y2 - y1)^2)
Using the given points (-2, 3) and (4, 1), we can plug in the values into the distance formula:
√((4 - (-2))^2 + (1 - 3)^2)
= √(6^2 + (-2)^2)
= √(36 + 4)
= √40
≈ 6.32 (rounded to the nearest tenth)
Therefore, the distance between the two points is approximately 6.32 units.
4. The diagonal of a rectangular prism can be found using the formula:
√(length^2 + width^2 + height^2)
Using the given measures of 3 cm, 4 cm, and 10 cm for length, width, and height respectively, we can plug in the values into the formula:
√(3^2 + 4^2 + 10^2)
= √(9 + 16 + 100)
= √125
≈ 11.2 (rounded to the nearest tenth)
Therefore, the diagonal of the rectangular prism is approximately 11.2 cm.
5. To find the height of the tent, we can use the Pythagorean theorem again. The height of the tent is the vertical distance from the apex (top) of the pyramid to the base. We can use the slant height and the base edge to find this height.
Let h represent the height of the tent.
Using the Pythagorean theorem, we have:
20^2 = h^2 + 13.5^2
400 = h^2 + 182.25
h^2 = 400 - 182.25
h^2 = 217.75
Taking the square root of both sides, we get:
h ≈ 14.8 (rounded to the nearest tenth)
Therefore, the height of the tent is approximately 14.8 meters.