can you find it?
Question. A geometry textbook has the following problem: “A right triangle has a hypotenuse equal to 10 and an altitude to its hypotenuse equal to 6. What is the area of the triangle?”
Can you figure out the correct answer?
solution:
.
.
.
The area of a triangle is given by the formula (base)(height)/2.
If the hypotenuse is taken as the base, and its altitude is taken as the height, then the area of the triangle would be (10)(6)/2 = 30. But this is not the correct answer!
The correct answer is there is no right triangle with the given dimensions. A right triangle with a hypotenuse of 10 can have an altitude on its hypotenuse of at most 5, so its maximal area would be 25.
Here is one way to see this. Draw a line segment with a length of 10. The endpoints are two of the vertices for the triangle. Where can a third point be drawn to form a right triangle?
The angle opposite the hypotenuse must be a right angle of 90 degrees. This means the two sides of the triangle must subtend a 180 degree angle in a circle. The hypotenuse must be the diameter of a circle, and the third point can be any point on the circle (except the endpoints of the hypotenuse).
The vertical distance from the third point to the hypotenuse is the altitude to the hypotenuse. This is largest when the third point is at the top or bottom of the circle, and the vertical distance is equal to the radius of the circle (half the length of the hypotenuse, which is the diameter of the circle).
Therefore, a right triangle with a hypotenuse of 10 can have an altitude on its hypotenuse of at most 5.
The correct answer is the textbook has a mistake because there is no triangle with the stated dimensions.
BY:Vikash Rahii
Question. A geometry textbook has the following problem: “A right triangle has a hypotenuse equal to 10 and an altitude to its hypotenuse equal to 6. What is the area of the triangle?”
Can you figure out the correct answer?
solution:
.
.
.
The area of a triangle is given by the formula (base)(height)/2.
If the hypotenuse is taken as the base, and its altitude is taken as the height, then the area of the triangle would be (10)(6)/2 = 30. But this is not the correct answer!
The correct answer is there is no right triangle with the given dimensions. A right triangle with a hypotenuse of 10 can have an altitude on its hypotenuse of at most 5, so its maximal area would be 25.
Here is one way to see this. Draw a line segment with a length of 10. The endpoints are two of the vertices for the triangle. Where can a third point be drawn to form a right triangle?
The angle opposite the hypotenuse must be a right angle of 90 degrees. This means the two sides of the triangle must subtend a 180 degree angle in a circle. The hypotenuse must be the diameter of a circle, and the third point can be any point on the circle (except the endpoints of the hypotenuse).
The vertical distance from the third point to the hypotenuse is the altitude to the hypotenuse. This is largest when the third point is at the top or bottom of the circle, and the vertical distance is equal to the radius of the circle (half the length of the hypotenuse, which is the diameter of the circle).
Therefore, a right triangle with a hypotenuse of 10 can have an altitude on its hypotenuse of at most 5.
The correct answer is the textbook has a mistake because there is no triangle with the stated dimensions.
BY:Vikash Rahii
Evil geometry problem
![Evil geometry problem]()
Reviewed by biharishayar
on
December 26, 2017
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