The perimeter of
△
J
K
L
is 200 centimeters.
JK = 68 centimeters
KL = 62 centimeters
Which statement is true about the interior angles of
△
J
K
L
?
A
∡
L
is the greatest of the interior angles
A ,
∡
L
, , is the greatest of the interior angles
B
∡
L
is the least of the interior angles
B ,
∡
L
, is the least of the interior angles
C
∡
K
is the least of the interior angles
C ,
∡
K
, is the least of the interior angles
D
∡
K
is the greatest of the interior angles
D ,
∡
K
, , is the greatest of the interior angles
Very confusing when printed in a vertical orientation. Need to print horizontally.
To determine which statement is true about the interior angles of triangle JKL, we need to use the given information.
The perimeter of triangle JKL is 200 centimeters, and the lengths of its sides are given as:
JK = 68 centimeters
KL = 62 centimeters
By definition, the perimeter of a triangle is the sum of the lengths of its sides. Therefore, we can find the length of the third side, JL, by subtracting the sum of the other two sides from the perimeter:
JL = Perimeter - JK - KL
JL = 200 - 68 - 62
JL = 200 - 130
JL = 70 centimeters
Now, let's analyze the possible statements:
A. ∡L is the greatest of the interior angles
B. ∡L is the least of the interior angles
C. ∡K is the least of the interior angles
D. ∡K is the greatest of the interior angles
To determine which statement is true, we can use the property that in any triangle, the sum of the measures of the interior angles is always equal to 180 degrees.
Let's assume that ∡J, ∡K, and ∡L represent the measures of angles J, K, and L, respectively.
Therefore, we have the following equations:
∡J + ∡K + ∡L = 180 (1)
∡J is the measure of angle J
∡K is the measure of angle K
∡L is the measure of angle L
We also know that JK + KL + JL is equal to the perimeter of the triangle:
JK + KL + JL = 200 (2)
JK = 68 (3)
KL = 62 (4)
JL = 70 (5)
Substituting equations (3), (4), and (5) into equation (2), we have:
68 + 62 + 70 = 200
200 = 200
This shows that our values are correct.
Now, using equation (1), we can solve for ∡L:
∡J + ∡K + ∡L = 180
∡L = 180 - ∡J - ∡K
Since the measures of angles J and K are not given, we cannot determine the specific values of ∡J and ∡K. Therefore, we cannot conclusively determine whether ∡L is the greatest or the least of the interior angles.
Hence, the answer is: None of the given statements (A, B, C, or D) about the interior angles of triangle JKL are true, as we do not have enough information to confirm any specific relationship between the angles.
To determine which statement is true about the interior angles of △JKL, you can use the fact that the sum of the interior angles in a triangle is always 180 degrees.
Let's start by finding the third interior angle, ∡J. Since the perimeter of △JKL is given as 200 centimeters, we know that JK + KL + JL (the third side) must equal 200.
Given:
JK = 68 centimeters
KL = 62 centimeters
To find JL, we can subtract the lengths of JK and KL from the perimeter:
JL = 200 - (JK + KL)
JL = 200 - (68 + 62)
JL = 200 - 130
JL = 70 centimeters
Now that we have the lengths of all three sides, we can use the Law of Cosines to find the cosine of each angle:
cos(∡J) = (JK² + JL² - KL²) / (2 * JK * JL)
cos(∡K) = (KL² + JK² - JL²) / (2 * KL * JK)
cos(∡L) = (JL² + KL² - JK²) / (2 * JL * KL)
Next, we can use the inverse cosine function (cos^(-1)) to find the measure of each angle (in degrees) using a calculator or software:
∡J = cos^(-1)((JK² + JL² - KL²) / (2 * JK * JL))
∡K = cos^(-1)((KL² + JK² - JL²) / (2 * KL * JK))
∡L = cos^(-1)((JL² + KL² - JK²) / (2 * JL * KL))
Now, compare the measures of ∡J, ∡K, and ∡L to determine which angle is the greatest and which is the smallest. The statement corresponding to the angle with the greatest measure is true.
To summarize:
- Calculate JL using the perimeter equation.
- Use the Law of Cosines to find the cosine of each angle.
- Apply the inverse cosine function to find the measure of each angle.
- Compare the measures of the angles to determine the greatest and smallest ones.