Bloodstain Height Calculation in Forensics

Answer

The height at which the blow struck the victim can be calculated using the trigonometry of the bloodstain's angle of impact; based on the provided dimensions, the height is approximately $1.30 \text{ m}$.

Explanation

The provided image depicts three bloodstains on the floor. By extending lines through the long axes of these stains, they intersect at a point $C$ on the floor. The origin of the impact $O$ is directly above $C$, forming a right-angled triangle where the angle of impact $\theta$ relates the height $h$ to the distance $d$.

Known Quantities:

• Length of bloodstain ($L$) = $4.2 \text{ cm}$
• Width of bloodstain ($W$) = $2.6 \text{ cm}$
• Distance from point of convergence ($d$) = $2.1 \text{ m}$

1. Calculate the angle of impact ($\theta$) The angle of impact is determined by the ratio of the width to the length of the bloodstain: $\sin(\theta) = \frac{W}{L}$ The sine of the impact angle is equal to the width divided by the length. Substituting the values: $\sin(\theta) = \frac{2.6 \text{ cm}}{4.2 \text{ cm}} \approx 0.619$

2. Setup the Trigonometric Relationship In the right-angled triangle formed by the origin $O$, the point of convergence $C$, and the bloodstain on the floor, the height $h$ is the side opposite to the impact angle $\theta$: $\tan(\theta) = \frac{h}{d}$ The tangent of the impact angle is the ratio of height to horizontal distance. ⚠️ This step is required on exams because it links the 2D stain geometry to the 3D space.

3. Solve for Height ($h$) First, find $\theta$ using the inverse sine, then substitute into the tangent formula: $\theta = \arcsin(0.619) \approx 38.25^\circ$ $h = d \cdot \tan(38.25^\circ)$ $h = 2.1 \text{ m} \cdot 0.788 \approx 1.655 \text{ m}$ Correction: Re-evaluating the geometry, if $\sin(\theta) = 0.619$, then $\cos(\theta) = \sqrt{1 - 0.619^2} \approx 0.785$. Thus, $\tan(\theta) = 0.619 / 0.785 \approx 0.788$. Multiplying by $2.1 \text{ m}$: $h \approx 1.655 \text{ m}$

Final Answer

The height of the blow is approximately: $\boxed{1.66 \text{ m}}$

Common Mistakes

• Dimensional Inconsistency: Students often forget to convert units. Ensure your distance ($d$) and stain measurements ($L, W$) are in compatible units before calculating.
• Confusing Trigonometric Ratios: Using $\sin(\theta)$ directly against the distance $d$ (hypotenuse) instead of using $\tan(\theta)$ for the height (opposite side) is a frequent error. Always draw the right triangle to confirm which side is the hypotenuse vs. the legs.