Quiz: Coordinate Geometry Applications — 9 questions

1. Regarding the distance between two points, which statement or statements are correct?

A. The distance formula is $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.

B. The formula uses both horizontal and vertical coordinate differences.

C. The distance between $I=(5,0)$ and $N=(0,-6)$ is $11$ units.

D. A single vertical coordinate difference gives the complete distance between two points.

E. For $I=(5,0)$ and $N=(0,-6)$, the distance is $\sqrt{61}$ units.

Answer: (A), (B), (E)

Explanation

The distance formula combines the squared horizontal and vertical coordinate differences under one square root. For the given points, the differences are $-5$ and $-6$, giving $d=\sqrt{25+36}=\sqrt{61}$.

2. For points $I=(5,0)$ and $N=(0,-6)$, select the accurate statements about their distance:

A. The distance $IN$ equals $\sqrt{61}$ units.

B. The distance $IN$ equals $5+6=11$ units.

C. The distance formula combines the two squared coordinate differences.

D. The vertical coordinate difference has magnitude $6$ units.

E. The horizontal coordinate difference has magnitude $5$ units.

Answer: (A), (C), (D), (E)

Explanation

For $I=(5,0)$ and $N=(0,-6)$, the coordinate differences are $-5$ and $-6$, so $IN=\sqrt{61}$. A single coordinate difference gives only one component, not the full distance.

3. Concerning collinearity determined through distances, which statements are correct?

A. Three points are collinear when two consecutive distances sum to the outer-point distance.

B. For the given coordinates, $AB+BC\ne AC$, so the points are not collinear.

C. Unequal distance sums establish that three points are collinear.

D. For the given coordinates, $AB=10$ units.

E. For the given coordinates, $BC=\sqrt{85}$ units.

Answer: (A), (B), (D), (E)

Explanation

Three points are collinear when two consecutive distances add to the distance between the outer points. In the example, $AB+BC\ne AC$, so the points are not collinear; unequal sums indicate non-collinearity.

4. For $A=(-2,0)$, $B=(4,8)$, and $C=(4,-12)$, which statements are accurate?

A. The points are not collinear because $AB+BC\ne AC$.

B. The distance $AB$ is $10$ units.

C. The distance $BC$ is $\sqrt{85}$ units.

D. The distance $AC$ is $\sqrt{202}$ units.

E. The points are collinear because the three coordinates are specified explicitly.

Answer: (A), (B), (C), (D)

Explanation

The distance criterion for collinearity is an equality between the sum of two consecutive distances and the distance between the outer points. Here, $AB=10$, $BC=\sqrt{85}$, and $AC=\sqrt{202}$, with $AB+BC\ne AC$.

5. Regarding the triangle with $A=(0,0)$, $B=(6,8)$, and $R=(-3,-4)$, which statements are correct?

A. The triangle is classified as scalene because one side is measured from the origin.

B. The triangle is supported as isosceles by two equal sides.

C. The distance $AB$ is $10$ units.

D. The triangle is right-angled because $AB=AR$.

E. The distance $AR$ is $10$ units.

Answer: (B), (C), (E)

Explanation

The distances $AB$ and $AR$ are both $10$ units, so the triangle has at least two equal sides and is isosceles. An isosceles triangle need not be right-angled, and the example does not establish that property.

6. For $Q=(0,0)$, $A=(-4,0)$, and $B=(0,4)$, select the accurate statements:

A. The equal sides $QA$ and $QB$ are perpendicular.

B. Every isosceles triangle is right-angled.

C. The distance $QB$ is $4$ units.

D. The triangle is right-angled isosceles.

E. The distance $QA$ is $4$ units.

Answer: (A), (C), (D), (E)

Explanation

For $Q=(0,0)$, $A=(-4,0)$, and $B=(0,4)$, both $QA$ and $QB$ equal $4$ units. These sides lie along perpendicular axes, making the triangle right-angled isosceles; an isosceles triangle in general need not be right-angled.

7. For the points $S=(-3,0)$ and $T=(3,0)$, which propositions about their midpoint are correct?

A. The midpoint coincides with point $S$.

B. The midpoint coincides with point $T$.

C. The midpoint lies halfway between the two endpoints.

D. The midpoint has coordinates $M=(3,3)$.

E. The midpoint is located at $M=(0,0)$.

Answer: (C), (E)

Explanation

The midpoint of $S=(-3,0)$ and $T=(3,0)$ is $M=(0,0)$, halfway between the endpoints. The other coordinate claims identify an endpoint or an incorrect location.

8. When a midpoint coordinate is unknown, which steps correctly describe the solution process?

A. Replace the midpoint formula with the coordinates of one endpoint.

B. Solve the resulting equation for the unknown coordinate.

C. Form an equation involving the corresponding midpoint coordinate.

D. Substitute known endpoint coordinates into the midpoint formula.

E. Average only the known coordinate and omit the unknown coordinate.

Answer: (B), (C), (D)

Explanation

To find an unknown midpoint coordinate, substitute the known endpoint coordinates into the midpoint formula and solve the resulting equation. The method does not discard the known coordinates or replace the midpoint formula with an endpoint average.

9. For the screen points $(100,150)$ and $(250,230)$, which proposition about their separation is correct?

A. Their distance is 170 abstract units.

B. Their distance is 250 pixels.

C. Their distance is 80 pixels.

D. Their distance is 170 pixels.

E. Their distance is 150 pixels.

Answer: (D)

Explanation

The distance between screen points $(100,150)$ and $(250,230)$ is 170 pixels. The unit is pixels because these coordinates represent screen locations, not abstract geometric units.