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Math Chapter 7 : Topic 1
Distance Formula
1 / 20
The distance formula is derived using which fundamental theorem?
Reasoning: The horizontal and vertical differences form a right triangle, so the Pythagorean Theorem is used.
2 / 20
If two points lie on the same horizontal line, the distance between them is:
Reasoning: When $y_{1}=y_{2}$, vertical distance is zero, so distance depends only on horizontal difference.
3 / 20
The distance between points $(3, 5)$ and $(3, - 2)$ is:
Reasoning: $\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$$\sqrt{\left(3-3\right)^{2}+\left(-2-5\right)^{2}}=\sqrt{0+49}=\sqrt{49}=7$
4 / 20
Which expression correctly represents the horizontal distance between two points?
Reasoning: Horizontal distance is the absolute difference of $x$-coordinates.
5 / 20
If $A(1,2)$ and $B(4,6)$, what is the length of $AB$?
Reasoning: $\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$$\sqrt{\left(4-1\right)^{2}+\left(6-2\right)^{2}}=\sqrt{9+16}=\sqrt{25}=5$
6 / 20
If the distance between two points is zero, then:
Reasoning: Distance is zero only when both points coincide.
7 / 20
Which of the following pairs has distance $\sqrt{13}$?
Reasoning: $\sqrt{\left(2\right)^{2}+\left(3\right)^{2}}=\sqrt{13}$
8 / 20
The distance formula is applicable in:
Reasoning: Formula works for all real coordinates.
9 / 20
If one point is at origin, the distance formula simplifies to:
Reasoning: Substituting $(0,0)$ gives simplified form.
10 / 20
Distance between $(- 2, - 3)$ and $(2, 1)$ is:
Reasoning: $\sqrt{\left(4\right)^{2}+(4)^{2}}=\sqrt{32}$
11 / 20
Which step is essential in deriving the distance formula?
Reasoning: Squaring removes negative values and fits Pythagoras.
12 / 20
If $A(x_{1}, y_{1})$ and $B(x_{2}, y_{2})$, then $AB^{2}$ equals:
Reasoning: Square of distance follows directly from Pythagorean theorem.
13 / 20
Distance between $(0,0)$ and $(5,12)$ is:
Reasoning: $\sqrt{\left(5\right)^{2}+(12)^{2}}=\sqrt{25+144}=\sqrt{169}=13$
14 / 20
Which pair represents equal distances from origin?
Reasoning: Distance depends on squares, so signs/order don’t matter.
15 / 20
Distance between $(a, b)$ and $(- a, - b)$ is:
Reasoning: $\sqrt{\left(2a\right)^{2}+\left(2b\right)^{2}}=2 \sqrt{a^{2}+b^{2}}$
16 / 20
If distance between $(x, 2)$ and $(4, 2)$ is $5$, then $x$ is:
Reasoning: $\left|x-4\right|=5 ⟹ x=9$ or $-1$
17 / 20
Which situation forms a right triangle in derivation?
Reasoning: Perpendicular projections create right triangle.
18 / 20
Distance between $(1,1)$ and $(4,5)$ equals:
Reasoning: $\sqrt{25}=5$
19 / 20
Which coordinate difference represents vertical side?
Reasoning: Vertical distance depends on $y$-coordinates.
20 / 20
If distance between two points is $\sqrt{50}$, simplified form is:
Reasoning: $\sqrt{50}=\sqrt{25\times 2 }$ $=5\sqrt{2}$
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