Right Triangles Review Worksheet
Geometry & Coordinate Algebra Advanced Review
Triangles & Distance Mastery
Name:
Date: Period:
1
Pythagorean Theorem
\(a^2 + b^2 = c^2\) • Converse: \(c^2 < a^2+b^2\) (acute), \(c^2 > a^2+b^2\) (obtuse)
2
Coordinate Distance
\(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} = \sqrt{\Delta x^2 + \Delta y^2}\)
3
Special Right Ratios
\(45^\circ\text{-}45^\circ\text{-}90^\circ \rightarrow x : x : x\sqrt{2}\) | \(30^\circ\text{-}60^\circ\text{-}90^\circ \rightarrow x : x\sqrt{3} : 2x\)
I
Precision Foundations & Exact Values
Simplify all radicals completely.
1. Unknown Leg with Radicals 3 pts
Right \(\triangle JKL\) has hypotenuse \(JL = 4\sqrt{6}\) and leg \(JK = 2\sqrt{10}\). Find the exact simplified length of leg \(KL\).
Exact \(KL =\)
2. Exact Distance & Midpoint 3 pts
For points \(P(-2, 7)\) and \(Q(6, -9)\), compute exact distance \(PQ\) and the midpoint \(M\).
\(d(P, Q) =\) Midpoint \(M =\)
II
Special Right Triangles & Composite Figures
R Q P S 60° 30° 45° 6
\(\triangle PQR\) is \(30^\circ\text{-}60^\circ\text{-}90^\circ\); \(\triangle PRS\) is \(45^\circ\text{-}45^\circ\text{-}90^\circ\) (\(PR\) hyp).
3. Multi-Step Perimeter Analysis 5 pts
In the figure, \(m\angle Q = 90^\circ\), \(m\angle P = 30^\circ\), and \(QR = 6\). \(\triangle PRS\) is an isosceles right triangle with hypotenuse \(PR\).
(a) Find exact lengths of \(QP\), \(PR\), and leg \(RS\).
(b) Express the exact perimeter of quadrilateral \(PQSR\).
\(QP =\)
\(RS = SP =\)
Perimeter \(PQSR =\)
III
Coordinate Plane Synthesis & Geometric Proof
6 pts
4. Coordinate Triangle Classification
Given vertices \(A(-3, 2)\), \(B(1, 5)\), and \(C(4, 1)\) on the coordinate plane:
Step 1: Side Lengths
Show radical calculations:
\(AB =\)
\(BC =\)
\(AC =\)
Step 2: Pythagorean Converse
Test \(a^2 + b^2\) vs \(c^2\):
Type:
Step 3: Angle Deduction
Deduce exact acute angles:
\(m\angle A =\) \(m\angle C =\)
High-Order Geometry Review • Page 1 of 2 Turn over for Parts IV - VI →
Part IV - VI
Higher-Order Modeling & Synthesis
Student Name:
IV
Spatial Geometry: 3D Space Diagonal
6 pts
l = 8 ft w = 9 ft h = 12 ft d Vault: \(8 \times 9 \times 12\text{ ft}\)
An equipment vault measures \(l = 8\text{ ft}\), \(w = 9\text{ ft}\), and \(h = 12\text{ ft}\). A rigid grounding rod connects opposite bottom and top corners.
(a) Two-Step Derivation: Show how 2D Pythagoras twice yields \(d = \sqrt{l^2+w^2+h^2}\):
(b) Calculate exact rod length: \(d =\) ft
V
Mathematical Critique & Error Analysis
5 pts
Problem Context
"In a \(30^\circ\text{-}60^\circ\text{-}90^\circ\) triangle, the hypotenuse is \(10\text{ cm}\). Find exact lengths of both legs."
× Student Solution (Jordan):
1. Ratio is \(x : x\sqrt{3} : 2x\).
2. Long Leg \(= 10\sqrt{3}\text{ cm}\) (multiplied by \(\sqrt{3}\)).
3. Short Leg \(= \frac{10\sqrt{3}}{2} = 5\sqrt{3}\text{ cm}\).
1. Identify the conceptual error:
2. Correct exact side lengths:
Short: Long:
VI
Non-Routine Challenge: The Asymmetrical Truss
7 pts
An asymmetrical roof truss \(T_1\text{-}V\text{-}T_2\) has vertical king-post \(VH = 12\text{ m}\) perpendicular to base tie-beam \(T_1 T_2\) at \(H\).
45° 30° T₁ V T₂ H h = 12 Left rafter: 45° • Right rafter: 30°
(a) Find exact lengths of left rafter \(VT_1\) and right rafter \(VT_2\).
(b) Find the exact total horizontal span \(T_1 T_2 = T_1 H + HT_2\).
(c) Express total truss perimeter plus strut length \((\text{Total Steel} = VT_1 + VT_2 + T_1 T_2 + VH)\) in simplest radical form.
Workspace (exact radical form):
\(VT_1 =\) , \(VT_2 =\)
Span \(T_1 T_2 =\)
Total Length \(=\)
High-Order Geometry Review • Page 2 of 2 Total Possible Points: 25 • Mastery Benchmark: ≥ 21/25
