Math without numbers

Geometry Level 3

A rectangle is constructed from $\color{#D61F06}m$ rows of $\color{#20A900} n$ squares, each with a side length of $\color{#EC7300} k$ . Which of the following statements must be true? Add the numbers that represent the true statements.

The perimeter of the rectangle is $\color{#69047E}2k(m+n)$ .
The area of the rectangle is $\color{#69047E}(kmn)^2$ .
There are ( $\color{#69047E}2m+2n$ ) unique squares along the perimeter of the rectangle.
The length of the diagonal of the rectangle is $\color{#69047E}k\sqrt{m^2+n^2}$ .
If $k$ is increased by a factor $\color{#3D99F6}c$ , then the area is increased by a factor of $\color{#69047E}c^2mn$ .

If you think the first three statements are true, enter your answer as $1+2+3=6\implies 6$ is the answer.

10 8 9 6 5 14

1 solution

J D
Jul 8, 2017

The perimeter of the rectangle is the sum of all the side lengths. So P = km + km + kn + kn, and we can factor out a k and a 2 to get 2k(n+m). So statement 1 is true.
The area of the rectangle is the product of the length and the width. The length is km and the with is kn, so A = km*kn = (k^2)(mn). This does not equal (kmn)^2 so statement 2 is false.
The number of unique rectangles along the perimeter is the sum of the number of rectangles on each side, minus the 4 corners (because we double count these rectangles since they appear on two sides). So m + m + n + n - 4 = 2(m+n-2). This does not equal 2(m+n), so statement 3 is false.
The length of the diagonal is the square root of the squares of the length and the width. So sqrt( (km)^2 + (kn)^2 ) = sqrt( (k^2)(m^2 + n^2) ) = k(sqrt( m^2 + n^2 ). So statement 4 is true.
If k is increased by a factor of c, then, since the area (see statement 2) is proportional to k^2, it will increase by a factor of c^2. This does not equal (c^2)mn, so statement 5 is false.

Only statements 1 and 4 were true, so we get 1+4=5 as our answer!

Nice and 100% true solution. I just need one favor from you, try surrounding your equations with latex brackets and see the difference.

Hana Wehbi - 3 years, 11 months ago

Math without numbers

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