Question

19.6 mi, 21.5 mi, 26.1 mi, 8.4 mi, p = ? mi, a = ? mi²

Explanation:

Step1: Calculate the Perimeter (P)

The perimeter of a triangle is the sum of all its sides. The sides are \( 21.5 \) mi, \( 26.1 \) mi, and \( 8.4 \) mi? Wait, no, wait. Wait, looking at the triangle, the sides are \( 21.5 \) mi, \( 26.1 \) mi, and wait, maybe I misread. Wait, the vertical side is \( 19.6 \) mi? No, wait, the triangle has sides: let's check again. Wait, the figure: there's a triangle with sides \( 21.5 \) mi, \( 26.1 \) mi, and \( 8.4 \) mi? Wait, no, maybe the base is \( 8.4 \) mi, and the other two sides are \( 21.5 \) and \( 26.1 \)? Wait, no, perimeter is sum of all sides. Wait, let's list the sides: \( 21.5 \) mi, \( 26.1 \) mi, and \( 8.4 \) mi? Wait, no, maybe I made a mistake. Wait, the problem: P is perimeter, so sum of all three sides. Wait, the sides are \( 21.5 \), \( 26.1 \), and \( 8.4 \)? Wait, no, wait, the vertical dashed line is \( 19.6 \) mi, but that's the height? Wait, no, maybe the triangle has sides: let's check the labels. The triangle has sides: \( 21.5 \) mi, \( 26.1 \) mi, and \( 8.4 \) mi? Wait, no, maybe the three sides are \( 21.5 \), \( 26.1 \), and \( 8.4 \)? Wait, no, that can't be. Wait, maybe the base is \( 8.4 \) mi, and the other two sides are \( 21.5 \) and \( 26.1 \)? Wait, let's calculate perimeter first. Perimeter \( P = 21.5 + 26.1 + 8.4 \). Let's compute that: \( 21.5 + 26.1 = 47.6 \), then \( 47.6 + 8.4 = 56 \) mi. Wait, is that right? Wait, maybe I misread the sides. Wait, the figure: there's a triangle with sides \( 21.5 \) mi, \( 26.1 \) mi, and \( 8.4 \) mi? Wait, maybe the vertical side is not a side but the height? Wait, no, the perimeter is sum of all sides. Let's check again. The problem says "P =? mi" and "A =? mi²". So perimeter is sum of the three sides: \( 21.5 + 26.1 + 8.4 \). Let's calculate that: \( 21.5 + 26.1 = 47.6 \), \( 47.6 + 8.4 = 56 \) mi. Now area: the area of a triangle is \( \frac{1}{2} \times base \times height \). The base here is \( 8.4 \) mi, and the height is \( 19.6 \) mi (the vertical dashed line). So area \( A = \frac{1}{2} \times 8.4 \times 19.6 \). Let's compute that: \( 8.4 \times 19.6 = 164.64 \), then \( \frac{164.64}{2} = 82.32 \) mi². Wait, is that correct? Let's verify the height. Wait, the height is the perpendicular distance from the base to the opposite vertex. The vertical dashed line is \( 19.6 \) mi, so that's the height. The base is \( 8.4 \) mi. So area is \( \frac{1}{2} \times 8.4 \times 19.6 \). Let's compute \( 8.4 \times 19.6 \): \( 8 \times 19.6 = 156.8 \), \( 0.4 \times 19.6 = 7.84 \), so total \( 156.8 + 7.84 = 164.64 \). Then half of that is \( 82.32 \) mi². And perimeter: sum of the three sides: \( 21.5 + 26.1 + 8.4 \). Let's add \( 21.5 + 8.4 = 29.9 \), then \( 29.9 + 26.1 = 56 \) mi. So that's the perimeter.

Step2: Calculate the Area (A)

Using the formula for the area of a triangle, \( A = \frac{1}{2} \times base \times height \). Here, base \( = 8.4 \) mi, height \( = 19.6 \) mi. So \( A = \frac{1}{2} \times 8.4 \times 19.6 \). As calculated before, \( 8.4 \times 19.6 = 164.64 \), then \( \frac{164.64}{2} = 82.32 \) mi².

Answer:

Perimeter \( P = 56 \) mi, Area \( A = 82.32 \) mi²