Question

1. when a line parallel to one side of a triangle divides the other two sides proportionally, which criterion can be used to prove the triangles are similar?

a. hl criterion
b. sas criterion
c. sss criterion
d. aa criterion

1. circle a has a radius of 3 and is centered at (0, 0). circle b has a radius of 9 and is centered at (5, 5). what transformations are needed to map circle a onto circle b?

a. reflection only
b. dilation only
c. rotation and translation
d. dilation and translation

1. in △pqr, ∠pqr = 90°, and qs is the altitude to pr. if pq = 6 and qr = 8, what is the length of qs if pr = 10?

a. 5.2
b. 2.8
c. 3.6
d. 4.4

1. what is the result of a dilation with a scale factor between 0 and 1?

a. the figure enlarges.
b. the figure disappears.
c. the figure stays the same size.
d. the figure is reduced in size.

1. what property of a figure does a dilation always preserve?

a. side lengths
b. area
c. volume
d. angles

1. if a rectangle has dimensions of 10 meters by 5 meters and is scaled by a factor of 2, what are the new dimensions?

a. 15 meters by 10 meters
b. 30 meters by 15 meters
c. 20 meters by 10 meters
d. 5 meters by 2.5 meters

Explanation:

Step1: Analyze question 22

The AA (Angle - Angle) criterion is used to prove similarity when a line parallel to one side of a triangle divides the other two sides proportionally. This is because the corresponding angles formed are equal.

Step2: Analyze question 23

To map Circle A (radius 3, center \((0,0)\)) to Circle B (radius 9, center \((5,5)\)), we first need a dilation (to change the radius from 3 to 9, scale factor \(k = 3\)) and then a translation (to move the center from \((0,0)\) to \((5,5)\)).

Step3: Analyze question 24

In right - triangle \(PQR\) with \(\angle PQR=90^{\circ}\), area of \(\triangle PQR=\frac{1}{2}\times PQ\times QR=\frac{1}{2}\times6\times8 = 24\). Also, area of \(\triangle PQR=\frac{1}{2}\times PR\times QS\). Given \(PR = 10\), then \(\frac{1}{2}\times10\times QS=24\), so \(QS=\frac{24\times2}{10}=4.8\). Wait, no, using the formula \(QS=\frac{PQ\times QR}{PR}\) (since area \(A=\frac{1}{2}PQ\times QR=\frac{1}{2}PR\times QS\)), \(QS=\frac{6\times8}{10}=4.8\). Wait, no, correct formula: In right - triangle \(PQR\) with altitude \(QS\) to hypotenuse \(PR\), \(QS=\frac{PQ\times QR}{PR}\). Substituting \(PQ = 6\), \(QR = 8\), \(PR = 10\), \(QS=\frac{6\times8}{10}=4.8\). Wait, no, \(PQ = 6\), \(QR = 8\), then by Pythagoras \(PR=\sqrt{6^{2}+8^{2}} = 10\). Area \(A=\frac{1}{2}\times6\times8=\frac{1}{2}\times10\times QS\), solving for \(QS\) gives \(QS=\frac{6\times8}{10}=4.8\). Wait, no, correct calculation: \(6\times8\div10 = 4.8\).

Step4: Analyze question 25

A dilation with a scale factor between 0 and 1 results in a reduction of the figure's size.

Step5: Analyze question 26

A dilation always preserves angles. Side lengths, area, and volume (for 3D) are changed by the scale factor.

Step6: Analyze question 27

If a rectangle has dimensions \(10\) meters by \(5\) meters and is scaled by a factor of \(2\), the new dimensions are \(10\times2 = 20\) meters and \(5\times2=10\) meters.

Answer:

1. d. AA Criterion
2. d. Dilation and translation
3. c. 4.8 (Note: There was a miscalculation in the options provided, but using the formula \(QS=\frac{PQ\times QR}{PR}\), with \(PQ = 6\), \(QR = 8\), \(PR = 10\), \(QS=\frac{6\times8}{10}=4.8\))
4. d. The figure is reduced in size
5. d. Angles
6. c. 20 meters by 10 meters