Question

1. what is the radius of the circle with the equation (x - 1)^2+(y - 4)^2 = 49?

a. 5
b. 4
c. 6
d. 7

1. what do you call the point where two line - segments meet in a polygon?

a. vertex
b. axis
c. midpoint
d. ray

1. if △abc has sides ab = 6 cm, ac = 8 cm, and ∠bac = 70°, which of the following triangles is congruent to △abc by sas?

a. △def with de = 6 cm, df = 8 cm, ∠edf = 75°
b. △def with de = 6 cm, df = 9 cm, ∠edf = 70°
c. △def with de = 7 cm, df = 8 cm, ∠edf = 70°
d. △def with de = 6 cm, df = 8 cm, ∠edf = 70°

Explanation:

Step1: Recall circle - equation formula

The standard form of a circle equation is $(x - a)^2+(y - b)^2=r^2$, where $(a,b)$ is the center and $r$ is the radius.

Step2: Identify radius from given equation

For the equation $(x - 1)^2+(y - 4)^2 = 49$, comparing with the standard form, we have $r^2=49$. Then $r=\sqrt{49}=7$.

Step3: Recall polygon - vertex definition

The point where two line - segments meet in a polygon is called a vertex.

Step4: Apply SAS congruence criterion

The Side - Angle - Side (SAS) congruence criterion states that two triangles are congruent if two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle. For $\triangle ABC$ with $AB = 6$ cm, $AC = 8$ cm, and $\angle BAC=70^{\circ}$, $\triangle DEF$ with $DE = 6$ cm, $DF = 8$ cm, and $\angle EDF = 70^{\circ}$ is congruent to $\triangle ABC$ by SAS.

Answer:

1. d. 7
2. a. Vertex
3. d. $\triangle DEF$ with $DE = 6$ cm, $DF = 8$ cm, $\angle EDF = 70^{\circ}$