Question

The coordinates of the vertices of △JKL are J(0, 2), K(3, 1), and L(1, −5). Drag and drop the choices into each box to correctly complete the sentences. The slope of JK¯¯¯¯¯ is_______, the slope of KL¯¯¯¯¯ is , and the slope of JL¯¯¯¯¯ is. △JKL is a right triangle because________.

a. Positive, negative, positive; it satisfies the Pythagorean theorem
b. Negative, positive, negative; it has one 90-degree angle
c. Positive, negative, negative; the sum of squares of the two shorter sides equals the square of the longest side
d. Negative, positive, positive; the sum of squares of the two shorter sides equals the square of the longest side

Answer 1

The slopes of the sides of triangle JKL are calculated as negative for JK, positive for KL, and negative for JL. Triangle JKL is not a right triangle because the sum of the squares of the two shorter sides does not equal the square of the longest side.

Step-by-step explanation:

The coordinates of the vertices of triangle JKL are J(0, 2), K(3, 1), and L(1, −5). To determine if triangle JKL is a right triangle, we must first find the slopes of the sides.

The slope of − − JK is calculated as (1-2)/(3-0) = -1/3, which is a negative slope. The slope of − − KL is (1+5)/(3-1) = 6/2 = 3, which is a positive slope. The slope of − − JL is (-5 - 2)/(1 - 0) = -7, which is also a negative slope.

Next, to confirm if triangle JKL is a right triangle, we apply the Pythagorean theorem. The length of JK can be calculated using the distance formula d = √((x2 − x1)2 + (y2 − y1)2) which gives √((3-0)2 + (1-2)2) = √(10). Similarly, for KL and JL, we get the lengths √(20) and √(49), respectively.

Checking the Pythagorean relationship a2 + b2 = c2, we have 10 + 20 = 30, which is not equal to 49, the square of the hypotenuse length. Therefore, △JKL is not a right triangle. The correct completion of the sentences is thus: The slope of JK−− is negative, the slope of KL−− is positive, and the slope of JL−− is negative. △JKL is not a right triangle because the sum of the squares of the two shorter sides does not equal the square of the longest side, which is option (d).