Triangles
Parallelograms
Rhombus, Square, & Rectangle
Trapezoid & Kite
Miscellaneous
100
What are the properties of an Isosceles triangle?
- all angles add up to 180 degrees - 2 congruent sides - Opposite angles of those sides are congruent
100
What are at least 2 properties of parallelograms?
- Opposite sides are parallel - Opposite sides are congruent - Angles add up to 360 degrees - Diagonals intersect at same point
100
True or False: In a rhombus, the diagonals intersect at the same point. If false, explain what shapes have diagonals that intersect at the same point.
What is FALSE. Only in a square and rectangle do the diagonals intersect at the same point.
100
What are the properties of an isosceles trapezoid?
- all angles add up to 360 degrees - one pair of parallel sides - non-parallel sides are congruent
100
State the midpoint formula and explain what it is used for.
M = (x2 + x1 / 2 , y2 + y1 / 2). Midpoint formula is used to find a point HALFWAY between two coordinate points.
200
Which set of numbers could be the lengths of the sides of a right triangle? (1) {10, 24, 26} (2) {12, 16, 30} (3) {3, 4, 6} (4) {4, 7, 8}
What is (1) {10, 24, 26}
200
What are the 3 methods to prove parallelograms? Explain why we use each.
- 4 distance to show 2 pair congruent sides - 4 slope to show 2 pair parallel sides - 2 midpoint on diagonals to show diagonals intersect
200
Before we prove that the shape is a rhombus, square, or rectangle, what must we prove first?
We must prove that the shape is a parallelogram before we can prove it is a rhombus, square, or rectangle.
200
What formula do we use to prove kites? Explain
We use 4 distances to show 2 pair of adjacent sides are congruent.
200
State the distance formula and explain what it is used for.
d = square root of (x2 - x1) ^2 + (y2 - y1) ^2. The distance formula is used to find the distance or length between two points.
300
What 2 formulas do we need to use in order to prove if a triangle is a right triangle or not? Explain how and why each formula is used.
We need to use the distance formula 3 times to find the length of each side of the triangle, and then substitute those lengths into the Pythagorean Theorem to see if it satisfies the equation. If it satisfies the Theorem, it is a right triangle.
300
Prove that the quadrilateral LMNO with the coordinates L(-2,3), M(4,3), N(2,-2) and O(-4,-2) is a parallelogram.
M LN=(0, 0.5); M MO=(0, 0.5) Therefore parallelogram; dLM=6 units; dNO=6 u; dMN=5.39u; dLO=5.39 u; mLM=0 u; mON=0 u; mMN=5/2 u; mLO=5/2 u
300
Prove a quadrilateral with vertices G(1,1), H(5,3), I(4,5) and J(0,3) is a rectangle.
Parallelogram: M GI=(2.5, 3); M HJ=(2.5, 3) Rectangle: dGI=5 u;dJH=5 u
300
What formulas do we use to prove trapezoids?
4 slope (one pair parallel - same slope and one pair not parallel – diff slope)
300
What is the difference between m and M?
m represents slope and M represents midpoint.
400
Prove ABC is a right triangle if A(1 in, 7 in) B(4 in, 7 in), C(4 in, 3 in)
dAB=3 inches; dBC= 4 inches; dAC=5 inches; 3^2 + 4^2 = 5^2
400
Romeo is surveying for a new parking lot shaped like a parallelogram. He knows that three of the vertices of the parallelogram ABCD are A(0, 0), B(5, 2), and C(6, 5). Find the coordinates of point D and justify mathematically.
D(1, 3). Slope of BC is 3 u, therefore slope of AD must also be 3 units. Starting from A(0, 0), going up 3 units and to the right 1 unit we get point (1, 3).
400
Prove that a quadrilateral with vertices A(-1,3), B(3,6), C(8,6) and D(4,3) is a rhombus.
dAD=5 u; dBC=5 u; dAB=5 u; dCD=5 u therefore ABCD is a rhombus because all fours sides are congruent.
400
Trapezoid RSUT, RS || TU, X is the midpoint of RT, V is the midpoint of SU. RS = 30 feet, XV = 44 feet, what is the length of TU?
Since X and V are the midpoints, XV is the midsegment of the trapezoid, and its length is equal to the average of RS and TU. (30 + x) / 2= 44; 30 + x = 88; x = 58 feet
400
Given parallelogram ABCD, angle B=5x, angle C=2x + 12, find the number of degrees of angle D.
x=24, 5(24) = 120 degrees
500
Prove that A (0, 1), B (3, 4), C (5, 2) is a right triangle.
dAB=4.24u, dBC=2.83 u; dAC=5.1u, 4.24^2 + 2.83^2 - 5.1^2 -> 26 u = 26 u Therefore ABC is a right triangle
500
In parallelogram ABCD, diagonals AC and DB intersect at E. AE = 3x - 4 and EC = x + 12. What is the value of x?
3x - 4 = x + 12; x = 8
500
Prove that the quadrilateral with vertices A(-1,0), B(3,3), C(6,-1) and D(2,-4) is a square.
Rhombus (4 sides equal): dAB=5u; dBC=5u; dCD=5u; dAD=5u. Rectangle (Congruent Diagonals) dAC=7.07u; dDB=7.07u
500
Prove that quadrilateral JACK with the vertices J(1, 3) A(-1, 1) C(-1, -2) and K(4, 3) is an isosceles trapezoid.
mAJ: 1-3 / -1-1 = 1; mCK: 3+2/4+1 = 1; mJK: 0/3; mAL: 3/0 undefined; Therefore Quad JACK is a trapezoid. Isosceles: dAC: 3 units dJK: 3 units; Quad JACK is an isosceles trapezoid because it is a trapezoid with 2 congruent sides.
500
You and a friend go hiking. You hike 3 miles north and 2 miles west. Starting from the same point, your friend hikes 4 miles east and 1 mile south. How far apart are you and your friend?
(-2, 3) and (4, -1); d=7.2 miles
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