Geometry Problem - Height of a Right Triangle

Given information:

• Height drawn from the vertex of the right angle in a right triangle is 6 cm.
• The height divides the hypotenuse into two segments, one of which measures 9 cm.

To find:

• The lengths of the sides of the triangle.

Solution: Let's denote the lengths of the two segments of the hypotenuse as x and y, with x being the segment adjacent to the height.

Using the Pythagorean theorem, we know that in a right triangle, the sum of the squares of the lengths of the two legs (sides adjacent to the right angle) is equal to the square of the length of the hypotenuse.

So, we have the following equation: x^2 + 6^2 = y^2

We also know that the height divides the hypotenuse into two segments, one measuring 9 cm. Therefore, we have the equation: x + y = 9

Now, we can solve these two equations simultaneously to find the values of x and y.

From the second equation, we can express x in terms of y: x = 9 - y

Substituting this value of x into the first equation, we get: (9 - y)^2 + 6^2 = y^2

Expanding and simplifying the equation: 81 - 18y + y^2 + 36 = y^2 117 - 18y = 0 18y = 117 y = 117/18 y ≈ 6.5 cm

Now, substituting the value of y back into the equation x = 9 - y, we get: x = 9 - 6.5 x ≈ 2.5 cm

Therefore, the lengths of the sides of the right triangle are approximately:

• The side adjacent to the height: x ≈ 2.5 cm
• The side opposite to the height: 6 cm
• The hypotenuse: y ≈ 6.5 cm