Practice Test 7 - Geometry

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. A right triangle $\triangle ABC$ is shown with the right angle marked at $A$. The altitude $AD$ is drawn to the hypotenuse $BC$ with $AD\perp BC$ marked at $D$. The legs are labeled $AB=b$ and $AC=a$, and the hypotenuse is labeled $BC=c$. The diagram is not drawn to scale. Which statement justifies the Pythagorean Theorem by using similarity rather than stating the formula?
2. A parabola is defined by focus $F(0,5)$ and directrix $y=1$ (shown on the coordinate plane). The parabola opens upward. Which reasoning correctly derives the equation?​
3. A rectangle has width $a$ and height $a$. A vertical strip of width $b$ is removed from the right side, leaving a rectangle of width $a-b$ and height $a$. That remaining rectangle is then partitioned by a horizontal segment into two rectangles of heights $a-b$ and $b$. Which polynomial identity applies here for all real $a$ and $b$?​
4. A coordinate geometry student claims that the circles $x^2 + y^2 = 4$ and $(x-6)^2 + (y-8)^2 = 36$ are not similar because their equations look completely different. How should this reasoning be evaluated?
5. Circle $\odot G$ and Circle $\odot H$ are shown with different radii. Their centers are marked at $G$ and $H$. Which reasoning uses similarity correctly to justify that the circles are similar?
6. A parabola is the set of points equidistant from focus $F(4,1)$ and directrix $x=0$. Which equation follows from the focus-directrix definition?
7. Segment $\overline{JK}$ is dilated about center $O$ to form $\overline{J'K'}$. The diagram shows $OJ=4$ units and $OJ'=10$ units. Which claim about length is supported by the diagram?
8. A right triangle has an angle $\theta$ with opposite side 4 units and adjacent side 3 units. Which value is $\theta$ to the nearest tenth of a degree?
9. A circle is the set of all points a fixed distance from its center. What is the equation of the circle with center $(2, -5)$ and radius $3$?​
10. A company models shipping boxes as rectangular prisms. A box with dimensions 12 inches by 9 inches by 6 inches costs $2.40 to ship. If shipping cost is proportional to volume, what would it cost to ship a box with dimensions 15 inches by 12 inches by 8 inches?
11. Two circles, $\odot R$ and $\odot U$, are drawn with different radii and with centers marked at $R$ and $U$. Which transformation maps $\odot R$ to a circle similar to $\odot U$ while keeping the idea of similarity (same shape) explicit?
12. In a proof that all circles are similar, a student writes: "Since all circles have the same shape, they are similar by definition." What is the primary flaw in this reasoning?
13. Which transformation is represented by the matrix $A=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$?
14. Consider the transformation $f(x,y) = (-y, x)$. Which statement best describes the properties of this transformation function?
15. Quadrilateral ABCD has vertices $A(0,1)$, $B(4,4)$, $C(7,0)$, and $D(3,-3)$ connected in that order. What is the perimeter of the figure?
16. Consider a sphere of radius $R$ and a cylinder of radius $R$ and height $2R$ that contains two congruent cones as follows: each cone has its base as the circle in the cylinder's mid-plane and its apex at one end of the cylinder (one apex at the top center, one at the bottom center). Slices are taken by planes parallel to the bases at the same height in all solids. Which statement justifies the sphere's volume formula using Cavalieri's principle (without relying only on memorized formulas)?
17. A linear transformation $T$ is represented by the matrix $A=\begin{pmatrix}-1&0\\0&-1\end{pmatrix}.$ On the coordinate plane, the triangle with vertices $(0,0)$, $(2,0)$, and $(0,1)$ is shown. Which statement describes the geometric effect of applying $T$ to the triangle?
18. In right triangle $\triangle JKL$, the right angle is at $K$. The hypotenuse is $JL=15$, and $\angle J = 50^\circ$. What is the length of leg $JK$?
19. Two circles are drawn on a coordinate plane: Circle $\odot P$ has center $P(-2,1)$ and Circle $\odot Q$ has center $Q(4,-1)$. Their radii are different, and the centers are marked. Which transformation maps $\odot P$ to a circle similar to $\odot Q$ using similarity transformations (not formulas)?
20. A non-right triangle $\triangle DEF$ is shown. Sides $DE=a$ and $DF=b$ form the included angle at $D$, labeled $C$. A dashed altitude from $E$ meets side $DF$ at a right angle. Which expression represents the area of the triangle?

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