A rectangle has length twice its width. A square has the same perimeter as the rectangle. Column A: The area of the rectangle / Column B: The area of the square

The relationship cannot be determined from the information given

In triangle $ABC$, $m\angle A=50^\circ$ and $m\angle B>m\angle C$. Column A: $m\angle B$ / Column B: $65^\circ$

The relationship cannot be determined from the information given

Study on the fly›ISEE Upper›Quantitative Reasoning › Plane geometry (polygons, triangles, angles)

Quantitative ReasoningFree

Plane geometry (polygons, triangles, angles) — ISEE Upper practice questions

Plane geometry (polygons, triangles, angles) on ISEE Upper Quantitative Reasoning covers rectangles, squares, triangles, and other polygons, including sides, perimeters, areas, and angle relationships. A student has to reason about how a change in length or width affects area, compare a rectangle with a square that shares a perimeter, and apply triangle facts without heavy calculation. Items appear as word problems or quantitative comparisons asking which quantity is larger. Common traps include mixing perimeter with area, treating a percent change in one side as the same change in area, and assuming equal perimeters mean equal areas. Answer choices are four labeled options, often with a cannot-be-determined choice on comparisons.

Section: Quantitative Reasoning 37 questions on the paper
179 practice questions in the app

Sample questions

Question 1

A rectangle's length is decreased by $20\%$ , and its width is increased by $50\%$ . The area of the new rectangle is what percent of the area of the original rectangle?

A $70\%$
B $100\%$
C $120\%$
D $130\%$

Show answer

Answer: C — $120\%$

The new length is 0.8 of the original and the new width is 1.5 of the original. The area scale factor is $0.8\times1.5=1.2$ , so the new area is $120\%$ of the original.

Question 2

A rectangle has length twice its width. A square has the same perimeter as the rectangle.

Column A	Column B
The area of the rectangle	The area of the square

AThe quantity in Column A is greater
BThe quantity in Column B is greater
CThe two quantities are equal
DThe relationship cannot be determined from the information given

Show answer

Answer: B — The quantity in Column B is greater

Let the rectangle's width be $w$ , so its length is $2w$ and its perimeter is $6w$ . The square's side length is $\dfrac{6w}{4}=\dfrac{3w}{2}$ . The rectangle's area is $2w^2$ , while the square's area is $\dfrac{9w^2}{4}$ , so Quantity B is greater.

Question 3

In triangle $ABC$ , $m\angle A=50^\circ$ and $m\angle B>m\angle C$ .

Column A	Column B
$m\angle B$	$65^\circ$

AThe quantity in Column A is greater
BThe quantity in Column B is greater
CThe two quantities are equal
DThe relationship cannot be determined from the information given

Show answer

Answer: A — The quantity in Column A is greater

Angles $B$ and $C$ sum to $180^\circ-50^\circ=130^\circ$ . If they were equal, each would be $65^\circ$ . Since $m\angle B>m\angle C$ , angle $B$ must be greater than $65^\circ$ , so Quantity A is greater.

Practice 179 Plane geometry (polygons, triangles, angles) questions in the app

Plane geometry (polygons, triangles, angles) — ISEE Upper practice questions

Sample questions

More ISEE Upper topics

Quantitative Reasoning