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
SAT Hardest Problems
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:
\[\begin{array} &\angle AOC=6x, &\angle BOE=x^2, &\angle DOF=y. \end{array}\]
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)\(\ \ 40\)
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,
\[\begin{align} \angle BOE + \angle EOC + \angle COA&=180^\circ \\ x^2+20+6x&=180\\ x^2+6x-160&=0\\ (x+16)(x-10)&=0\\ x&=10. \qquad (\text{since } x \text{ is a positive number}) \end{align}\]
Therefore, \(x+y=10+20=30\) and the correct answer is (C).
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)\(\ \ 14\)
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 \[\begin{align} \angle ABC+\angle BAF+(\angle CAF+\angle ACF)+\angle BCF&=180^\circ \\ 26^\circ+2x^\circ+9x^\circ+3x^\circ&=180^\circ \\ 14x^\circ&=154^\circ \\ x&=11. \end{align}\]
Therefore, the correct answer is (B).
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 \[\begin{array} &\lvert\overline{AB}\rvert=m, &\lvert\overline{AC}\rvert=n, &\lvert\overline{BC}\rvert=p. \end{array}\] 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)\(\ \ \frac{\left(m^2+n^2\right)\pi}{8}\)
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 \[\begin{align} \frac{\pi}{2} \left(\frac{m}{2}\right)^2+\frac{\pi}{2} \left(\frac{n}{2}\right)^2+\frac{m\times n}{2} -\frac{\pi}{2} \left(\frac{p}{2}\right)^2 &=\frac{\pi}{8}(m^2+n^2-p^2)+\frac{mn}{2}\\ &=\frac{mn}{2}. \quad (\text{since }m^2+n^2=p^2 \text{ by } (1)) \end{align}\]
Therefore, the correct answer is (C).
Incorrect Choices:
(A), (B), (D), and (E)
The solution explains why these choices are wrong.
SAT Tips for Geometry
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 \(\frac{x}{360} \cdot \pi r^2.\)
SAT General Tips