Analytic Geometry — Triangles & Quadrilaterals
Analytic Geometry — Triangles & Quadrilaterals. Practice questions to deepen understanding of triangles and quadrilaterals in analytic geometry. Online math practice with full solutions and step-by-step explanations.
Triangles and quadrilaterals — identifying parallelograms, rectangles, squares, and trapezoids from coordinates. Checking collinearity and slopes. 1️⃣ Triangles: do three points form a triangle? (slope check → collinearity); classifying triangles by slope; altitudes, medians, and special segments (basics only). 2️⃣ Quadrilaterals: parallelograms and more.
What type of triangle has vertices \(A(0,0)\), \(B(4,0)\), \(C(4,3)\)?
AB = 4, AC = 5, BC = 3 → Pythagorean theorem → right triangle.
Find the midpoint of side AB in a triangle where \(A(2,6)\), \(B(8,2)\).
Midpoint: \(\left(\frac{2+8}{2},\frac{6+2}{2}\right)=(5,4)\).
Do the points \(A(0,0)\), \(B(4,0)\), \(C(6,3)\), \(D(2,3)\) form a parallelogram?
AB = CD in the same direction, BC = AD in the same direction → parallelogram.
In a quadrilateral: \(A(1,1)\), \(C(7,5)\). What is the midpoint of diagonal AC?
Midpoint: (1+7)/2, (1+5)/2 = (4,3).
Is the quadrilateral with vertices \(A(0,0)\), \(B(4,0)\), \(C(4,3)\), \(D(0,3)\) a rectangle?
Two pairs of parallel sides + right angles.
Which of the points is closest to \(C(4,4)\)? A(3,4) B(4,1) D(1,1)
A is closest because the distance is 1.
Is the quadrilateral with points \(A(0,0)\), \(B(4,0)\), \(C(3,3)\), \(D(1,3)\) a trapezoid?
AB is parallel to CD → trapezoid.
What is the length of the diagonal of a rectangle with sides 6 and 8?
Pythagorean theorem: \(\sqrt{6^2+8^2}=10\).
In the parallelogram ABCD shown, which side is perpendicular to the x-axis?
A side perpendicular to the x-axis is vertical — where x values of both points are the same. Side CD has the same x value for both points, so it is perpendicular to the x-axis.
In right triangle \(A(0,0)\), \(B(6,0)\), \(C(0,8)\). What is the midpoint of the hypotenuse?
The hypotenuse is between B and C. Midpoint: (6+0)/2, (0+8)/2 = (3,4).
Given quadrilateral ABCD: A(1,1), B(5,1), C(6,4), D(2,4). Is ABCD a parallelogram?
In a parallelogram — the diagonals bisect each other. Midpoint of AC = (3.5,2.5), Midpoint of BD = (3,2.5). Not equal → not a parallelogram.
In quadrilateral A(0,0), B(4,2), C(6,2), D(2,0). Is AB ∥ CD?
Slope of AB = 2/4 = 0.5. Slope of CD = (2-2)/(6-2) = 0. Different slopes → not parallel.
In quadrilateral A(0,0), B(4,2), C(6,2), D(2,0). Is AB ∥ CD?
Slope of AB = 2/4 = 0.5. Slope of CD = (2-2)/(6-2) = 0. Different slopes → not parallel.
In trapezoid A(0,0), B(6,0), C(4,3), D(2,3). What is the distance from B to C?
BC = \(\sqrt{(4-6)^2 + (3-0)^2} = \sqrt{4+9} = \sqrt{13} \approx 3.6\).
In quadrilateral A(1,1), B(5,1), C(5,4), D(1,4). Is it a rectangle?
AB is horizontal, BC is vertical → 90° angles → rectangle.
Trapezoid ABCD with A(0,0), B(6,0), C(5,3), D(1,3). Find the midpoints of the diagonals.
AC: (0+5)/2, (0+3)/2 = (2.5,1.5) BD: (6+1)/2, (0+3)/2 = (3,1.5).
Find the distance from A(2,1) to C(7,5) in a quadrilateral.
\(\sqrt{(7-2)^2 + (5-1)^2} = \sqrt{25+16} = \sqrt{41}\).
In rectangle A(1,1), B(7,1), C(7,5), D(1,5). Is M(4,3) the center of the rectangle?
Midpoint of AC = (4,3). Midpoint of BD = (4,3). This is the center.
In parallelogram A(0,0), B(4,1), C(7,5), D(3,4). What is the length of diagonal AC?
\(\sqrt{(7-0)^2 + (5-0)^2} = \sqrt{49+25} = \sqrt{74}\).
A(0,0), B(3,0), C(3,3), D(0,3). What type of quadrilateral is this?
All sides equal (3), and all right angles → square.
Which point is closest to \(P(3,3)\)? A(1,5) B(4,1) C(6,7)
Distance to B: \(\sqrt{(4-3)^2 + (1-3)^2} = \sqrt{1+4} = \sqrt{5}\), the smallest.
Given quadrilateral ABCD: \(A(1,1)\), \(B(5,2)\), \(C(7,6)\), \(D(3,5)\). What type of quadrilateral is it?
Check midpoints of diagonals: midpoint AC: \(\left(\frac{1+7}{2},\frac{1+6}{2}\right)=(4,3.5)\) midpoint BD: \(\left(\frac{5+3}{2},\frac{2+5}{2}\right)=(4,3.5)\) Equal midpoints → parallelogram.
Given points \(A(1,2)\), \(B(5,3)\), \(C(8,7)\). Find point D so that ABCD is a parallelogram.
In a parallelogram: \(\vec{AD} = \vec{BC}\) or \(\vec{AB} = \vec{DC}\). \(\vec{BC} = (8-5,7-3) = (3,4)\). Therefore \(D = A + \vec{BC} = (1+3, 2+4) = (4,6)\).
Given quadrilateral with vertices \(A(0,0)\), \(B(4,0)\), \(C(5,3)\), \(D(1,3)\). Which statement is correct?
AB ∥ CD, and the diagonal sides AD and BC are equal. This is exactly the shape of an isosceles trapezoid (wide base, shorter upper base).
Quadrilateral ABCD: \(A(0,0)\), \(B(4,0)\), \(C(6,k)\), \(D(2,k)\). For what value of k will the quadrilateral be a rhombus?
AB = 4. Check AD: \(\sqrt{(2-0)^2 + (k-0)^2} = \sqrt{4+k^2}\). Require AD = 4 → \(4+k^2 = 16 \Rightarrow k^2 = 12 \Rightarrow k = \pm\sqrt{12}\).
In triangle ABC: \(A(0,0)\), \(B(6,0)\), \(C(4,4)\). M is the midpoint of AB and N is the midpoint of AC. What is the length of MN?
Calculate: \(M(3,0)\), \(N(2,2)\). Distance MN: \(\sqrt{(3-2)^2 + (0-2)^2} = \sqrt{1+4} = \sqrt{5} \approx 2.24\).
In parallelogram A(1,2), B(5,3), C(7,7), D(3,6). Find the intersection point of the diagonals.
In a parallelogram the diagonals bisect each other, so it is enough to find the midpoint of one diagonal: midpoint of AC: \(\left(\frac{1+7}{2},\frac{2+7}{2}\right) = (4, 4.5)\).
Quadrilateral with vertices \(A(0,0)\), \(B(3,1)\), \(C(4,4)\), \(D(1,3)\). Which statement is true?
Calculate side lengths: AB: \(\sqrt{10}\) BC: \(\sqrt{10}\) CD: \(\sqrt{10}\) DA: \(\sqrt{10}\). All sides equal → rhombus.
In quadrilateral A(0,0), B(6,0), C(5,3), D(1,3). Which statement is correct?
AB ∥ CD because both have the same y. Calculate the legs: AD: \(\sqrt{10}\) BC: \(\sqrt{10}\). Equal legs → isosceles trapezoid.
Triangle ABC: \(A(0,0)\), \(B(4,0)\), \(C(k,3)\). For what value of k is the triangle isosceles with AC = BC?
AC: \(\sqrt{k^2+9}\) BC: \(\sqrt{(k-4)^2+9}\). Require AC = BC → \(k^2 = (k-4)^2 \Rightarrow k = 2\).
Quadrilateral ABCD: \(A(1,1)\), \(B(5,1)\), \(C(6,4)\), \(D(2,4)\). What is ABCD?
AB = 4, BC = \(\sqrt{10}\), CD = 4, DA = \(\sqrt{10}\). Two pairs of equal opposite sides and right angles → rectangle (but not a square since AB ≠ BC).
Quadrilateral ABCD is shown in the diagram. Which side is perpendicular to the x-axis?
A side perpendicular to the x-axis is a vertical side — where the x value of both points is the same. In the diagram: B and C share the same x value, so BC is perpendicular to the x-axis.
In rectangle PQRS shown. Which side is perpendicular to the x-axis?
A vertical side is one where the x value is constant. Points Q and R share the same x value, so QR is the side perpendicular to the x-axis.
In the diagram, quadrilateral MNPQ is shown. Which side is perpendicular to the x-axis?
Side NP is vertical because the x value of N and P is the same, so it is perpendicular to the x-axis.
In trapezoid XYZW, which side is perpendicular to the x-axis?
WX is perpendicular because both points have the same constant x value (x = 100), making it the vertical side.