Question

1. what is the relationship between the centroid and the vertices of a triangle? a. it is two - thirds of the distance from each vertex to the midpoint of the opposite side b. it is equidistant from all vertices c. it forms right angles with the base of the triangle d. it lies on the circumcircle 5. if △abc = △def by asa, and ab = 5 cm, ∠a = 60°, and ∠b = 50°, what must be true? a. ∠d = 60° and ∠e = 50° b. side de = 4 cm c. ∠f = 50° d. side df = 6 cm 6. which of the following real - world applications demonstrates sas similarity? a. ensuring proportionality in triangular architectural designs b. comparing the heights of two poles based on triangular shadows c. measuring the radius of a circular path d. calculating the area of a field

Explanation:

1. The centroid of a triangle is the point of intersection of its medians and it is two - thirds of the distance from each vertex to the mid - point of the opposite side.
2. If $\triangle ABC\cong\triangle DEF$ by ASA (Angle - Side - Angle) congruence criterion, corresponding angles are equal. So if $\angle A = 60^{\circ}$ and $\angle B=50^{\circ}$ in $\triangle ABC$, then $\angle D = 60^{\circ}$ and $\angle E = 50^{\circ}$ in $\triangle DEF$.
3. SAS (Side - Angle - Side) similarity is about the proportionality of two sides and the included angle in two triangles. Ensuring proportionality in triangular architectural designs uses SAS similarity.

Answer:

1. a. It is two - thirds of the distance from each vertex to the midpoint of the opposite side
2. a. $\angle D = 60^{\circ}$ and $\angle E = 50^{\circ}$
3. a. Ensuring proportionality in triangular architectural designs