Question

if mnop is a rhombus, find mp.
what steps would you take to solve the problem above?
first,
then,
scroll more
drag these tiles and drop them in the correct blank above
use the pythagorean theorem
9x - 77 + 3x + 7 = 180
9x - 77 = 3x + 7
substitute into 3x + 7 only
9x - 77 + 3x + 7 = 90
substitute into either expression

Explanation:

Step1: Recall Rhombus Diagonals Property

In a rhombus, the diagonals bisect each other and the sides are equal. Wait, actually, in a rhombus, the diagonals are perpendicular bisectors, but also, the triangles formed are congruent. Wait, more importantly, in a rhombus, the diagonals bisect the angles, but also, the sides are equal. Wait, no, the key here is that in a rhombus, the diagonals bisect each other, but also, the triangles formed by the diagonals are congruent. Wait, actually, in a rhombus, the diagonals are perpendicular bisectors, so the triangles are right triangles. But first, we know that in a rhombus, the diagonals bisect each other, so the segments of the diagonals are equal? Wait, no, the sides of the rhombus are equal. Wait, looking at the diagram, \( PN \) and \( ON \)? Wait, no, the sides \( PN \) and \( MN \)? Wait, the labels are \( P \), \( N \), \( O \), \( M \)? Wait, the rhombus is \( MNOP \)? Wait, the sides: \( PM \) and \( ON \)? Wait, the expressions are \( 9x - 77 \) and \( 3x + 7 \). In a rhombus, all sides are equal. So \( PM = ON \)? Wait, no, maybe \( PN = MN \)? Wait, no, the problem is about finding \( MP \), and the sides are \( 9x - 77 \) and \( 3x + 7 \). So first, in a rhombus, all sides are equal, so \( 9x - 77 = 3x + 7 \). Wait, that's the first step: set the two side expressions equal because in a rhombus, all sides are equal. So the first step is \( 9x - 77 = 3x + 7 \). Then, solve for \( x \), then substitute into either expression to find the length.

Wait, the tiles: the first step should be setting the two side lengths equal because in a rhombus, all sides are equal. So the first tile is \( 9x - 77 = 3x + 7 \). Then, the next step would be solving for \( x \), then substituting into either \( 3x + 7 \) or \( 9x - 77 \) to find the length, then maybe using the Pythagorean theorem? Wait, no, maybe the diagonals? Wait, the problem is to find \( MP \), which is a side? Wait, no, \( MP \) is a side? Wait, the rhombus is \( MNOP \), so \( MP \) is a side. So first, since all sides of a rhombus are equal, set \( 9x - 77 = 3x + 7 \). So the first step is \( 9x - 77 = 3x + 7 \). Then, solve for \( x \):

Step1: Set Side Lengths Equal (Rhombus Property)

In a rhombus, all sides are equal. So \( 9x - 77 = 3x + 7 \).

Step2: Solve for \( x \)

Subtract \( 3x \) from both sides: \( 9x - 3x - 77 = 7 \) → \( 6x - 77 = 7 \).
Add 77 to both sides: \( 6x = 7 + 77 \) → \( 6x = 84 \).
Divide by 6: \( x = 14 \).

Step3: Substitute \( x \) into Side Expression

Substitute \( x = 14 \) into \( 3x + 7 \): \( 3(14) + 7 = 42 + 7 = 49 \). Or into \( 9x - 77 \): \( 9(14) - 77 = 126 - 77 = 49 \). So the side length is 49. But the tiles: the first step is \( 9x - 77 = 3x + 7 \), then solve for \( x \), then substitute.

But the tiles given are: "use the Pythagorean Theorem", "9x - 77 + 3x + 7 = 180", "9x - 77 = 3x + 7", "substitute into either expression", "substitute into 3x + 7 only".

So first, the correct first step is \( 9x - 77 = 3x + 7 \) because in a rhombus, all sides are equal. Then, the next step is solving for \( x \), then substituting into either expression. So the first tile to drag to "First" is \( 9x - 77 = 3x + 7 \), then "Then" would be "substitute into either expression" or solving, but the tiles include \( 9x - 77 + 3x + 7 = 180 \) (which is for a parallelogram with consecutive angles supplementary, but a rhombus is a parallelogram, but consecutive angles are supplementary, but the sides are equal, so that's not the case here). Wait, maybe I made a mistake. Wait, the rhombus: maybe the diagonals? Wait, no, the sides are equal. So the first step is setting the two side lengths equal: \( 9x - 77 = 3x + 7 \). Then, solve for \( x \), then substitute. So the first tile is \( 9x - 77 = 3x + 7 \), then the next is "substitute into either expression".

So the first step (First) is \( 9x - 77 = 3x + 7 \), then (Then) is "substitute into either expression".

Answer:

First: \( 9x - 77 = 3x + 7 \)
Then: substitute into either expression