Question

38 which answer is the point where all 3 medians intersect on a triangle called.
a) orthocenter
b) incenter
c) circumcenter
d) centroid

39 which is the point where 3 altitudes intersect on a triangle.
a) circumcenter
b) orthocenter
c) incenter
d) centroid

40 find the value of x
(figure of a parallelogram with angles marked, and sides labeled (1/2)x + 15 and 2x + 15)

41 find the area of the quadrilateral.
(figure of a quadrilateral with dimensions 8.5, 6.5, 3.5, 6, 6.5)

Explanation:

Question 38

The centroid of a triangle is defined as the point where the three medians (lines from each vertex to the midpoint of the opposite side) intersect. The orthocenter is where altitudes intersect, the incenter is where angle bisectors intersect, and the circumcenter is where perpendicular bisectors intersect. So the correct answer is the one related to medians' intersection, which is centroid.

The orthocenter of a triangle is the point where the three altitudes (perpendicular segments from each vertex to the opposite side) intersect. The circumcenter is for perpendicular bisectors, incenter for angle bisectors, and centroid for medians. So the point of intersection of altitudes is the orthocenter.

Step1: Recall parallelogram angle property

In a parallelogram, consecutive angles are supplementary (sum to \(180^\circ\)). So \(\frac{1}{2}x + 15 + 2x + 15 = 180\).

Step2: Combine like terms

\(\frac{1}{2}x+2x + 15 + 15=180\) → \(\frac{5}{2}x+30 = 180\).

Step3: Subtract 30 from both sides

\(\frac{5}{2}x=180 - 30=150\).

Step4: Solve for x

Multiply both sides by \(\frac{2}{5}\): \(x = 150\times\frac{2}{5}=60\).

Answer:

D) CENTROID

Question 39