The area of rectangle

The rectangle consists of eight congruent triangles, of area $A$ . We are told that $6A=42$ so $A=7$ and $8A=\boxed{56}$ .

Since the diagonals of the rectangle split the whole rectangle into 4 equal parts, the shaded area is 3 quarters of the whole area.

$3A = 42$ $A = 14$ $4A = 56$

Let the length of the rectangle be $l$ and the width be $w$ . Then the area of the white triangle is $\frac{1}{2}*l*\frac{w}{2} = \frac{lw}{4}$ .

The red region has area $lw-\frac{lw}{4}=\frac{3}{4}lw=42 \implies lw = \boxed{56}$ .

Note: This problem has been changed by the Brilliant staff, my solution below is no longer accurate. Please see @Otto Bretscher 's solution.

Note that the height of one of the obtuse-angled triangles (call it $h_T$ ) is half the height of the rectangle (call it $h_R$ ). Furthermore, the bases of the obtuse-angled triangles are shared with the rectangle. Let $A_R$ be the area of the rectangle and $A_T$ be the area of one of the obtuse-angled triangles. Then $\begin{aligned} A_T &= \frac{1}{2}bh_T \\ &= \frac{1}{2}b\cdot\frac{1}{2}h_R \\ &=\frac{1}{4} A_R \end{aligned}$

Thus, $\begin{aligned} A_R-A_T &= 12.75 \\ A_R - \frac{1}{4} A_R &= \frac{51}{4} \\ \frac{3A_R}{4} &= \frac{51}{4} \\ A_R &= \boxed{17} \end{aligned}$

It’s 42 divided by 3 plus 42 = 56

The diagonals divide the rectangle into 4 equal triangles.....hence each of the triangle of tha shaded part has the same area....

42÷3=14....14×4=56

Let the rectangle have dimensions axb. The area of the left and right shaded triangles are (1/2)(a/2)(b/2). The area of the lower triangle is (1/2)(b/2)(a/2). The sum of the three triangles is (3/4)ab = 42, so ab = 56.

4 triangles are the same area. 3 of them are 42. So each is 14. So area=14*4=56

42 divided by its occupied volume which is 75% gives the total volume of 56

If in the rectangle we named h= high and b= base, the total surface of the rectangle will be = bh

The sum of the area of the three triangles is 3A= 1/2h(b/2) + 1/2h(b/2) + 1/2b(h/2) = 3/4bh, and i f 3A=42 then 3A=3/4bh = 42, then bh = 4/3 x 42 = 56. And bh is the total surface of the rectangle.

If each triangle is x, and 3x = 42, then x = 42/3 so, x = 14. we need 4x, therefore, 4x = 56

I just gested I’m a nood

The rectangle is divided in half TWICE by lines connecting at opposite corners. This creates 4 triangles which must all have the same area (although the triangles are not identical in shape) 3/4 of the triangles are shaded in which equals an area of 42. Now we just solve this problem.

Step 1: 3/4 = 42/x ( multiply 3 × X + 42 ×4 )

Step 2: 3x = 168 ( divide both sides by 3 )

Answer: x = 56

The rectangle divides into 4 equal triangles. 3 of the triangles have an area of 42. The solution is to find the area of the 4th triangle and add it to the 42. This sum will give you the total area of the rectangle.

So, since they are equal, Area of the 4th triangle is → 3A = 42; Therefore, A = 42/3 = 14 So, the area of the 4th triangle is 14 (which is the area of each triangle) Therefore, total area of rectangle is 14 × 4 or 42 + 14 = 56!

So the area of 3 triangles is 42.

The area of all three triangles is the same, hence the fourth one will have the same area too.

Divide 42 by 3

42/3 = 14

Hence the area of the unshaded region is 14.

Add the area to the area of 3 triangles

14 + 42 = 56

Hey there is a very simple way to do this.

First the rectangle is divided into 4 parts which are all congruent and equal.

Out of these 3 parts of the rectangle are also congruent and equal and all these 3 triangle's area is 42. So, the area of 1 Triangle = 42/3 = 14

Now, As we can see clearly that 1 Triangle area is 14 but the to complete the triangle we need 4 triangles.

So, we can simply multiply it by 4 = 14 * 4 = 56

It is very simple to do if you use this unitary method

Two diagonals of a rectangle divide it into four triangles which are equal. If we know that the area of 3 of these triangles is 42 then the area of 1 triangle is 42/3 which is 14. Since this rectangle is made of 4 equal triangles which have the area of 14 the area of the rectangle is 4 • 14= 56.

Easy as there are 3 shaded triangles and have an area of 42 . 42÷ 3 is 14 that means 4 triangles is 56 which is the whole rectangle logic people logic