SAT Geometry Perfect Score

To get a perfect score on SAT Math, you need to:

• Get every single problem correct.
• Have complete mastery of all of the SAT skills
• Remember the Tips and use them
• Figure out your common mistakes and avoid them

73415P1

Three line segments $\overline{AB}, \overline{CD}$ and $\overline{EF}$ intersect at point $O.$ The measures (in degrees) of some angles are as follows:

$$\begin{array}{c}&\angle AOC=6x, &\angle BOE=x^2, &\angle DOF=y. \end{array}$$

If the measures (in degrees) of $\angle OHG$ and $\angle OGH$ are $2y$ and $6y,$ respectively, what is $x+y?$

(A)$\ \ 20$
(B)$\ \ 25$
(C)$\ \ 30$
(D)$\ \ 35$
(E)$\ \ 40$

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Correct Answer: C

Solution:

Since the interior angles of a triangle sum to $180^\circ,$

$$y+2y+6y=180 \Rightarrow y=20.$$

Then since $\angle COE$ and $\angle DOF$ are vertical angles, $\angle COE=20^\circ.$ Hence,

$$\begin{aligned} \angle BOE + \angle EOC + \angle COA&=180^\circ \\ x^2+20+6x&=180\\ x^2+6x-160&=0\\ (x+16)(x-10)&=0\\ x&=10. \qquad (\text{since } x \text{ is a positive number}) \end{aligned}$$

Therefore, $x+y=10+20=30$ and the correct answer is (C).

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Incorrect Choices:

(A), (B), (D), and (E)
The solution explains why these choices are wrong.

73551wiki

What is $x$ in the above diagram?

Note: The above diagram is not drawn to scale.

(A)$\ \ 10$
(B)$\ \ 11$
(C)$\ \ 12$
(D)$\ \ 13$
(E)$\ \ 14$

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Correct Answer: B

Solution:

73551wikisolution

Draw a line segment connecting $A$ and $C,$ as shown in the figure above. Then you will find that $\angle EFC$ is an exterior angle of both $\triangle DEF$ and $\triangle AFC.$ This implies $$\angle EFC=\angle DEF+\angle EDF=x^\circ+8x^\circ=9x^\circ=\angle CAF+\angle ACF.$$ Since the interior angles of $\triangle ABC$ sum to $180^\circ,$ we have $$\begin{aligned} \angle ABC+\angle BAF+(\angle CAF+\angle ACF)+\angle BCF&=180^\circ \\ 26^\circ+2x^\circ+9x^\circ+3x^\circ&=180^\circ \\ 14x^\circ&=154^\circ \\ x&=11. \end{aligned}$$

Therefore, the correct answer is (B).

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Incorrect Choices:

(A), (C), (D), and (E)
The solution explains why these choices are wrong.

73710wiki

As shown in the above diagram, triangle $ABC$ is a right triangle with side lengths $$\begin{array}{c}&\lvert\overline{AB}\rvert=m, &\lvert\overline{AC}\rvert=n, &\lvert\overline{BC}\rvert=p. \end{array}$$ If the sides of $\triangle ABC$ are the diameters of their corresponding semicircles in the diagram, what is the area of the shaded region?

(A)$\ \ \frac{m^2+n^2}{4}$
(B)$\ \ \frac{mn}{4}$
(C)$\ \ \frac{mn}{2}$
(D)$\ \ \frac{mn\pi}{8}$
(E)$\ \ \frac{\left(m^2+n^2\right)\pi}{8}$

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Correct Answer: C

Solution:

By the Pythagorean theorem, we have $m^2+n^2=p^2. \qquad (1)$

Now, observe that the area of the shaded region can be obtained as follows: $$(\text{Area of semicircle with diameter } m)+(\text{Area of semicircle with diameter } n)\\+(\text{Area of } \triangle ABC)-(\text{Area of semicircle with diameter } p).$$ Then the area of the shaded region is $$\begin{aligned} \frac{\pi}{2} \left(\frac{m}{2}\right)^2+\frac{\pi}{2} \left(\frac{n}{2}\right)^2+\frac{m\times n}{2} -\frac{\pi}{2} \left(\frac{p}{2}\right)^2 &=\frac{\pi}{8}(m^2+n^2-p^2)+\frac{mn}{2}\\ &=\frac{mn}{2}. \quad (\text{since }m^2+n^2=p^2 \text{ by } (1)) \end{aligned}$$

Therefore, the correct answer is (C).

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Incorrect Choices:

(A), (B), (D), and (E)
The solution explains why these choices are wrong.

Lines and Angles

• Angles at a point sum to $360^\circ.$
• Angles on a line sum to $180^\circ.$
• $\angle A$ and $\angle B$ are complementary if $m\angle A + m\angle B=90^\circ.$
• $\angle A$ and $\angle B$ are supplementary if $m\angle A + m\angle B=180^\circ.$
• Vertical angles are congruent.
• The angle bisector divides an angle in half.
• The midpoint of a segment divides it in half.
• If a diagram is drawn to scale, trust it.

Parallel Lines

• Know the Properties of Parallel Lines.
• Angles on a line sum to $180^\circ.$
• $\angle A$ and $\angle B$ are complementary if $m\angle A + m\angle B=90^\circ.$
• $\angle A$ and $\angle B$ are supplementary if $m\angle A + m\angle B=180^\circ.$
• Vertical angles are congruent.
• The angle bisector divides an angle in half.
• Angles in a triangle sum to $180^\circ.$
• The two acute angles in a right triangle are complementary.
• An exterior angle in a triangle equals the sum of the two nonadjacent interior angles.
• If a diagram is drawn to scale, trust it.

Triangles

• The angles opposite the two congruent sides in an isosceles triangle are congruent.
• The measures of the angles in a triangle add to $180^\circ.$
• The measure of an exterior angle of a triangle equals the sum of the measures of the two non-adjacent interior angles.
• If one side of a triangle is longer than another side, then the angle opposite the first side is bigger than the angle opposite the second side.
• If one angle in a triangle is bigger than another angle, then the side opposite the first angle is longer than the side opposite the second angle.
• Triangle Inequality: The sum of the lengths of any two sides in a triangle is greater than the length of its third side.
• Perimeter of a polygon equals the sum of the lengths of its sides.
• Area of a triangle with height $h$ and base $b$: $A_{\triangle} = \frac{1}{2}bh.$
• If two triangles have equal heights, then the ratio of their areas equals the ratio of their bases.
• If two triangles have equal bases, then the ratio of their areas equals the ratio of their heights.
• If two figures are similar, and their scale factor is $a:b,$ then the ratio of their perimeters is $a:b$ and the ratio of their areas is $a^2:b^2.$

Right Triangles

• Pythagorean Theorem: $a^2 + b^2 = c^2.$
• If $c^2 = a^2 + b^2,$ then $m\angle C = 90$ and $\triangle ABC$ is right.
• If $c^2 < a^2 + b^2,$ then $m\angle C < 90$ and $\triangle ABC$ is acute.
• If $c^2 > a^2 + b^2,$ then $m\angle C > 90$ and $\triangle ABC$ is obtuse.
• Know the $30^\circ-60^\circ-90^\circ$ and the $45^\circ-45^\circ-90^\circ$ Theorems.
• AA Postulate: Two triangles are similar if two angles of one triangle are congruent to two angles of the other triangle.
• The measures of the angles in a triangle add to $180^\circ.$
• Perimeter of a polygon equals the sum of the lengths of its sides.

Polygons

• Know the Properties of Parallelograms.
• $A_{parallelogram} = bh,$ where $b$ is the length of the base, and $h$ is the height.
• Area of a triangle with height $h$ and base $b$: $A_{\triangle} = \frac{1}{2}bh.$
• Area of a square with side length $s: A_{\square} = s^2.$
• The sum of the measures of the interior angles of a convex polygon with $n$ sides is $180(n-2).$
• The sum of the measures of the exterior angles, one per vertex, of any convex polygon is $360^\circ.$

Circles

• The circumference of a circle with radius $r$ and diameter $d: C = 2\pi r = \pi d.$
• Area of a circle with radius $r: A_{\bigodot} = \pi r^2.$
• The measure of an arc equals the measure of its central angle.
• The length of an arc with measure $x$ is $\frac{x}{360}\cdot 2 \pi r.$
• The area of the sector formed by an arc measuring $x$ and two radii is $\frac{x}{360} \cdot \pi r^2.$

Solid Geometry

• Area of a triangle with height $h$ and base $b$: $A_{\triangle} = \frac{1}{2}bh.$
• Know the $30^\circ-60^\circ-90^\circ$ and the $45^\circ-45^\circ-90^\circ$ Theorems.
• Area of a circle with radius $r: A_{\bigodot} = \pi r^2.$
• The perimeter of a square with side length $s$: $P_{\square} = 4s.$
• The volume of a cube with edge length $s$: $V = s^3.$
• The volume of a rectangular solid with length $l,$ width $w,$ and height $h: V = l \cdot w \cdot h.$
• The surface area of a cube with edge length $s$: $SA = 6s^2.$
• Volume of a cylinder with base radius $r$ and height $h: V = \pi r^2 h.$

Composite Figures

• Area of a triangle with height $h$ and base $b$: $A_{\triangle} = \frac{1}{2}bh.$
• Know the $30^\circ-60^\circ-90^\circ$ and the $45^\circ-45^\circ-90^\circ$ Theorems.
• The perimeter of a square with side length $s$: $P_{\square} = 4s.$
• Area of a square with side length $s: A_{\square} = s^2.$
• Area of a rectangle with length $l$ and width $w: A = l\cdot w.$
• The volume of a cube with edge length $s$: $V = s^3.$
• The volume of a rectangular solid with length $l,$ width $w,$ and height $h: V = l\cdot w \cdot h.$
• The surface area of a cube with edge length $s$: $SA = 6s^2.$
• Volume of a cylinder with base radius $r$ and height $h: V = \pi r^2 h.$
• The circumference of a circle with radius $r$ and diameter $d: C = 2\pi r = \pi d.$
• Area of a circle with radius $r: A_{\bigodot} = \pi r^2.$
• The measure of an arc equals the measure of its central angle.
• The length of an arc with measure $x^\circ$ is $\frac{x}{360}\cdot 2 \pi r.$
• The area of the sector formed by an arc measuring $x$ and two radii is $\frac{x}{360} \cdot \pi r^2.$

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