Question

find the following measurements of the triangle shown.
base = \boxed{ } units
height = \boxed{?} units

Explanation:

Step1: Identify the base

Looking at the triangle, the side labeled 80 is the base (since the height is drawn to it, forming a right angle with the 40 side and the 50 side is part of the right triangle formed). So base = 80 units.

Step2: Use Pythagorean theorem for height

Wait, actually, the right triangle has legs 40 and height \( h \), hypotenuse 50? Wait no, wait the dashed line is height, 40 is one leg, 50 is hypotenuse? Wait no, maybe the base is 80? Wait no, let's re-examine. The triangle has a height (dashed) of 40? No, wait the right triangle: legs are 40 and height \( h \), hypotenuse 50? Wait no, maybe the base is 80, and the height is calculated? Wait no, the problem is to find base and height. Wait the base is 80 (the side with length 80), and the height is the dashed line? Wait no, the right triangle: 40, 50, and height? Wait 40 - 50 - 30? Wait no, \( 40^2 + h^2 = 50^2 \)? Wait \( 1600 + h^2 = 2500 \), so \( h^2 = 900 \), \( h = 30 \)? Wait no, maybe the base is 80, and the height is 30? Wait no, the base is 80 (the side with length 80), and the height is the altitude to that base. Wait the right triangle: 40, 50, and 30 (since 3-4-5 triangle scaled by 10: 30-40-50). So the height is 30, and the base is 80? Wait the problem says "base = | units" and "height =? units". Wait the base is 80 (the side labeled 80), and the height is 30? Wait no, let's check again. The triangle has a side of 80 (base), a side of 40 (one leg of the right triangle), and the hypotenuse of the right triangle is 50? Wait no, 40, 50, and 30 (since 30-40-50 is a right triangle, 3-4-5 scaled by 10). So the height is 30, and the base is 80? Wait the base is 80 units.

Wait the first part: base = 80 units (the side with length 80), and then height: using the right triangle with legs 40 and height \( h \), hypotenuse 50? Wait no, 40, 50, and 30: \( 30^2 + 40^2 = 900 + 1600 = 2500 = 50^2 \), so that's a right triangle. So the height is 30 units. Wait but the problem says "base = | units" – maybe the base is 80, and height is 30? Wait let's confirm. The triangle has a base of 80 (the side labeled 80), and the height is the altitude to that base, which is 30 (from the right triangle 30-40-50). So base is 80, height is 30.

Wait the first box is base: 80, then height: 30.

Step1: Determine the base

The base of the triangle is the side with length 80 units (as it's the side to which the height is drawn, forming a right angle with the 40 - unit side and the 50 - unit hypotenuse of the right triangle). So base = 80.

Step2: Calculate the height

Using the Pythagorean theorem in the right triangle with hypotenuse 50 and one leg 40. Let the height be \( h \). Then \( h^2+40^2 = 50^2 \).
\[

$$\begin{align*} h^2&=50^2 - 40^2\\ h^2&=2500 - 1600\\ h^2&=900\\ h&=\sqrt{900}\\ h&= 30 \end{align*}$$

\]

Answer:

base = 80 units, height = 30 units