The following are some of the multiple questions from the June 2019 New York State Geometry Regents exam.

### June 2019 Geometry, Part III

Each correct answer is worth up to 4 credits. Partial credit is available. Work must be shown. Correct answers without work receive only 1 point.

**32.** *Riley plotted A(-1,6), B(3,8), C(6,-1), and D(1,0) to form a quadrilateral.
Prove that Riley’s quadrilateral ABCD is a trapezoid.
[The use of the set of axes on the next page is optional.]
*

*Riley defines an isosceles trapezoid as a trapezoid with congruent diagonals. Use Riley’s definition
to prove that ABCD is not an isosceles trapezoid.
*

**Answer:**

Graphing the quadrilateral may help, so there isn't a reason not to. You already know that it is a trapezoid, so if you make a mistake on the graph, it should be obvious enough to fix. Also, graphing it will tell you which two slopes to check.

A trapezoid has one pair of parallel sides, and parallel sides have the same slope. You can also show that the other two sides are not parallel, with different slopes, but this is not necessary.

AD: (6 - 0) / (-1 - 1) = 6/-2 = -3

BC: (-1 -8) / (6 - 3) = -9/3 = -3 AD || BC.

Riley's quadrilateral is a trapezoid because it has a pair of parallel sides.

Next, use the distance formula to find the length of the diagonals. If they are the same, then it is an isosceles trapezoid. If there are not the same length, the trapezoid is not isosceles.

AC: sqrt ( (-1 - 6)^{2} + (6 - -1)^{2}) = sqrt(49 + 49) = sqrt(98)

BD: sqrt ( (8 - 0)^{2} + (3 - 1)^{2}) = sqrt(64 + 4) = sqrt (68)

AC =/= BD, so the trapezoid is not isosceles.

**33.** *A child-sized swimming pool can be modeled by a cylinder. The pool has a diameter of 6 1/2 feet and a height of 12 inches. The pool is filled with water to 2/3 of its height. Determine and state the volume of the water in the pool, to the nearest cubic foot.
*

*One cubic foot equals 7.48 gallons of water. Determine and state, to the nearest gallon, the number
of gallons of water in the pool.
*

**Answer:**

Half of 6.5 feet diameter is 3.25 feet radius. Twelve inches is 1 foot, and 2/3 of that is 2/3 feet.

V = pi * r^{2} * h = (3.141592)(3.25)^{2}(2/3) = 22.122

The pool holds about 22 cubic feet of water

22 * 7.48 = 164.56 = 165 gallons of water.

**34.** *4 Nick wanted to determine the length of one blade of the windmill pictured below. He stood at a
point on the ground 440 feet from the windmill’s base. Using surveyor’s tools, Nick measured the
angle between the ground and the highest point reached by the top blade and found it was 38.8°.
He also measured the angle between the ground and the lowest point of the top blade, and found
it was 30°.
*

Determine and state a blade’s length, x, to the nearest foot.

Determine and state a blade’s length, x, to the nearest foot.

**Answer:**

There are two right triangles. You need to find the opposite side (the height) of each of them. The difference between the two is the height of the blade, *x*.

You know the adjacent, 440, and you are looking for the opposite, so you need to use tan for both.

tan (30) = y / 440

y = 440 * tan (30) = 254.034...

tan (38.8) = z / 440

y = 440 * tan (38.8) = 353.769.--

353.8 - 254.0 = 99.8 = 100 feet.

### Part IV

A correct answer is worth up to 6 credits. Partial credit is available.

**35.** * Quadrilateral MATH, HM = AT, HT = AM, HE is perpendicular to MEA, and HA is perpendicular to AT
*

*Prove TA * HA = HE * TH
*

**Answer:**

I don't remember another time that they asked you to prove something like this.

Work backward for a step here.

If TA * HA = HE * TH then TA / TH = HE / HA.

In other words, sides are proportional. So you can show that two triangles are proportional, which will get you here. Two triangles will have proportional sides if they are similar, and similar triangles have three pairs of congruent angles, but you only have to find two.

Statements | Reasons |

1. Quadrilateral MATH, HM = AT, HT = AM, HE is perpendicular to MEA, and HA is perpendicular to AT | 1. Given |

2. Angle HEA and angle TAH are right angles. | 2. Perpendicular lines form right angles |

3. Angle HEA = Angle TAH | 3. All right angles are congruent. |

4. MATH is a parallelogram | 4. A quadrilateral with two pairs of congruent sides is a parallelogram. |

5. MA || TH | 5. Opposite sides of a parallelogram are parallel. (Definition of parallelogram) |

6. Angle THA = Angle EAH | 6. Alternate interior angles. |

7. Triangle HEA ~ Triangle TAH | 7. AA Theorem |

8. TA / TH = HE / HA | 8. Corresponding sides of similar triangles are proportional. |

9. TA * HA = HE * TH | 9. In a proportion, the cross-products are equal. |

**End of Exam**

How did you do?

Comments and corrections welcome. *(I get many of the latter!)*

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